Your question actually is related to:
Original Copies of Muhammad al-Bukhari's Books?
Can any hadith truly be called sahih anymore?
The first question is looking for an original manuscript of Sahih al-Bukhari. Which is no more existent while the second asks whether we can call hadith Sahih because once the books where compiled everybody referenced to them.
How hadith was and is transmitted until today
First of all you should be aware that hadith students today still try to get a (preferably high) sanad of hadith compilations this means they memorize the hadith with its Sanad (narrator chain) till the prophet () himself.
High sanad means a reduced amount of narrators between the student and the prophet() this would actually mean a narrator chain of around 27-29 narrators. This is the same procedure how qur'an was transmitted and is protected in the memory of people until today.
So just the fact that these compilations where fixed in a book which was accessible to a handful of people doesn't mean that oral transmission of the content ended and therefore a cross check between book content and memorization was always possible.
One must be aware that at the time books where very expensive and most students of knowledge have only their memory to rely on instead of money to buy books. Therefore only libraries with a good financial support would have copies of the books as a reference for scholars and students, while the later may have written their own manuscripts if they had the means to purchase paper/papyrus.
For example the first chapter of this article covers the Sanad of the contemporary Yemani sheikh Habib al-Jafri till imam al-Bukhari:
the bold letters cover the chain of the copy referred to as that of ibn al-Waqt later in my answer.
Note that among the narrators is mentioned ibn Hajar al-'Asqalani (14) and Zakriya ibn Muhammad al-Anasari (13).
1قراءة عن شيخه العارف الحبيب عبد القادر بن أحمد السقاف ،
2 وهو يرويه قراءة عن أبيه الإمام أحمد بن عبد الرحمن السقاف ،
3 الذي يرويه قراءة عن شيخه الإمام علي بن محمد الحبشي الذي يرويه ،
4 عن شيخه الإمام عيدروس بن عمر الحبشي ، وهو مسند حضرموت وله ثبت اسمه “منحة الفتاح الفاطر في أسانيد السادة الأكابر” ، ذكر فيه أسانيده في مروياته ،
5 عن شيخه الإمام عبد الله بن أحمد باسودان قال :
6 أرويه عن شيخنا السيد الإمام عمر بن عبد الرحمن البار ،
7 عن شيخنا وشيخه السيد البقية حامد بن عمر بن حامد آل أبي علوي،
8 عن شيخه الحبيب عبد الرحمن بن عبد الله بلفقيه ،
9 عن شيخه المسند الحسن بن علي العُجيمي والشيخ أحمد بن محمد المثلي ،
10 عن شيخهما حافظ عصره محمد بن علاء الدين البابلي ،
11 وهو يرويه عن أبي النجا سالم بن محمد السمهوري ،
12 عن خاتمة الحفاظ النجم محمد بن أحمد الغيطي ،
13 عن شيخ الإسلام زكريا بن محمد الأنصاري Zakariyya al-Ansari،
14 عن أمير المؤمنين في الحديث الحافظ أحمد بن علي بن حجر العسقلاني Ibn Hajar al-'Asqalani،
15 عن شيخه إبراهيم بن أحمد التنوخي وعبد الرحيم بن رزين الحموي ،
16 عن أبي الفضل أحمد بن أبي طالب الحجار ،
17 عن الحسين المبارك الزبيدي
18 عن أبي الوقت عبد الأول بن عيسى الهروي
19 ، عن أبي الحسن عبد الرحمن بن المظفر الداودي
20 ، عن أبي محمد عبد الله بن أحمد السرخسي As-Sarakhsi
21 ، عن أبي عبد الله بن محمد بن يوسف بن مطر الفربري Al-Frarbari
22 ، عن أمير المؤمنين في الحديث أبي عبد الله محمد بن إسماعيل البخاري رحمه الله تعالى
So between him and al-Bukhari are 22 Narrators (at least) so the highest sanad of a hadith here could reach 27 narrators in between Ali al-Jafri and the prophet ().
Among the books which cover those people who have transmitted hadith compilation we find the book of ibn Nuqtah ابن نقطة called at-Taqyeed li Ma'rifat as-Sunan wal-Masaneed التقييد لمعرفة رواة السنن والمسانيد (literally: The registartion to know the narrators of the Sunan and Musnads) this book can be regarded as a compendium of all narrators of hadith compilations (books: The two Sahihs, the Sunan of abi Dawod, ibn Majah, an-Nasa-i and Jami' at-Tirmidhi -this was the inquiry of a scholar from Qurtubah (Cordoba)- and he added beyond those scholars which he regarded as essential which cover narrators of almost all known books of Sunni hadith according his introduction) from the compilation of the book to the year ibn Nuqtah's book was completed (the book is quoting the names of scholars in alphabetical order). Ibn Nuqtah actually died 629 a.H. and his book reached Baghdad around 623 a.H..
How Sahih al-Bukhari was transmitted
Students of al-Bukhari and some of their students
Please note that Sahih al-Bukhari was compiled or completed around 232 a.H. -taking the statement reporetd by al-U'qayly as true which says that al-Bukhari presneted his book to the imams 'Ali ibn al-Madini (died 234 a.H.), Yahya ibn Maiy'n (died 233 A.h.) and Ahmad ibn Hanbal (died 241 a.H.)- while the author died 256 a.H..
Secondly Sahih al-Bukhari was transmitted via many of his students and narrators the most known of these are five:
Please note that unless otherwise stated any narrator introduced here was regarded as trustworthy and reliable especially in his narration of the Sahih.
- Muhammad ibn Yusuf Al-Farbari محمد بن يوسف الفربري died 320 a.H. ... I don't want to say that almost all printed copies of Sahih al-Bukhari are based on his narration and invite you to read my research results till the end. He almost was a newborn when al-Bukhari finished the compilation of his Sahih and lived long, so that he was able to share his knowledge long enough. His version of the Sahih was transmitted by at least seven known transmitters -according Arabic wikipedia- which I'll present later.
- Ibraheem ibn Ma'qil an-Nasafi إبراهيم بن معقل died 295 a.H. he was a qadi an author of a Tafsir and a hadith collection called al-Musnad al-Kabir (which must have been a huge book according to some reports about it). The most prominent narrator of his version of the Sahih was Khalaf ibn Muhammad al-Khiyam from Bukhara (died 361 a.H.) also known as abu Salih, he is regarded as a da'if narrator. Among those who heard Sahih al-Bukhari from abu Salih were the hafidh al-Khattabi أبو سليمان الخطابي (one of the earliest commentators on Sahih al-Bukhari, he also heard it from al-Frabari) and al-Hakim from Nishapur. Abu Salih was known as the narrator of Bukhara, he didn't travel for gaining knowledge, but had gathered a huge amount in Bukhara and was regarded as a hafidh of hadith. (For details see also here and the following pages in a book about the narrators of the Sahih)
- Hammaad ibn Shaakir حماد بن شاكر died 311 a.H. (see also here and in the following pages)and was a student of both al-Bukhari and abu 'Isa a-Thirmidhi. His narration was transmitted via at least two known students: Bakr ibn Muhammad ibn Ja'afar بكر بن محمد بن جعفر (Al-Hafidh Ja'afar ibn al-Mua'taz al-Mustaghfiri جعفر المستغفري said he heard the Sahih from him on 370 a.H. and 380 a.H.) and Ahamd ibn Muhamamd ibn Rameeh an-Nasawi أحمد بن محمد بن رميح النسوي died 357 a.H. and started teaching the Sahih 350 a.H, for 3 years in Nishapur according al-Hakim from Nishapur. Note that according hufadh like an Nawawi and ibn Salah his version has around 200 less content than that of other narrators of the Sahih al-Bukhari especially al-Farbari.
- Al-Bazdawi أبو طلحة منصور البزدوي (died 329 a.H.) he was the last of al-Bukhari's student who is known to have transmitted the Sahih. His version was copied by some local authorities, however scholars and hufadh regarded it as da'if due to his youth while hearing from al-Bukhari. The copy of his manuscript shows huge similarities to the version of of Hammad ibn Shaakir. (For details read here)
- Al-Mahaamili الحسين بن إسماعيل المحاملي (born 235 a.H., died 330 a.H.) he started his studies 244 a.H. imam a-Dhahabi quoted in his siyar a narration via al-Mahaamili from Sahih al-Bukahri, this is a way trough which we know that he is one of the direct students and transmitters of al-Bukahris hadith, but there's a doubt on whether he transmitted the whole Sahih or not. That's why al-hafidh ibn Hajar said after quoting al-Bazdawi that there's whom transmitted from the book of al-Bukari after him, but he didn't transmit the whole book and whoever claimed that al-Mahaamili was transmitting the Sahih made a huge mistake. As for his narrators we have 'Abdulwahid ibn Muhammad ibn Mahdi عبد الواحد بن محمد بن مهدي via whom a-Dhahhabi narrated the hadith in his Siyar (For further details see here and in the following pages), as for the second narrator he was referred to by ibn al-Bayyi'a ابن البيع (in Lessan al-Mizaan), while al-Hakim from Nishapur is known by this reference which is seemingly a mistake, but it can be corrected if we read in the commentary of al-Kiramani شمس الدين الكرماني on Sahih al-Bukhari where he mentioned his third riwaya of the sahih (the two first were sourced back to al-Farabri) to Abu Muhammad 'Abdullah ibn 'Ubaydallah ibn Yahya ibn Zakariya al-Mu'adib أبو محمد عبد الله بن عبيد الله بن يحي بن زكريا المؤدب who was also known by ابن البيع (See here).
As you may conclude from the above all scholars who deal with hadith sciences especially with Sahih al-Bukhari have at least one (connected) chain of transmission between them and the author and the author has a chain up to the prophet (). And you will find those commenting it showing slight linguistic differences in it or in other ahadith between different sources (narrators). Further the most reliable source from a subjective point of view seems al-Farbari, before Ibraheem ibn Ma'qil and Hammad ibn Shakir, while seemingly the later has more reliable students the former has more content.
Students of al-Farbari and some of their students
Now let's quote the well-known narrators/students of al-Farbari, before quoting some other narrations of Sahih al-Bukhari (I'll start with the 7 in the order of Arabic wikipedia):
Abu Ishaaq al-Mustamly إبراهيم بن أحمد المستمليّ died 376 a.H. and heard the Sahih 314 a.H. at least four hufadh are known among his students: Abu Dhar al-Harawi أبو ذر الهروي (355 a.H. - 434 a.H.) who was a teacher of many famous scholars such as ibn 'Abd al-Barr, al-Baji, a-Daany and al-Khateeb al-Baghdadi and many more. Ibn al-Kharaaz also known as عبد الرحمن بن عبد الله بن خالد الهمداني الوهراني (338 a.H. - 411 a.H.), al-Balkhy أحمد بن محمد بن العباس البلخي and abu 'Abdillah Muhammad ibn Ahamd ibn Muhammad an-Najjar أبو عبد الله محمد بن أحمد ابن محمد النجار. Note that he and some of his mates copied from al-Farbari's manuscript, however they found it incomplete therefore there's a difference in wording or order between them in these empty positions. (For details see here and in the following pages)
Abu Muhammad al-Hammuwi (293 a.H.- 381 a.H.) heard the Sahih from al-Farbari 316 a.H. and the book of a-Darimi from 'Isa ibn Umar ibn al-'Abbas as-Samarqandi, among his students are abu Dhar al-Harawi and others.
Abu al-Haytham al-Kushmihany محمد بن مكي بن محمد بن مكي بن زُراع الكشميهني (389 a.H.) among his students is again abu Dhar al-Harawi and the female scholar Karima al-Marwazziyya (365 a.H. - 463 a.H.) who was a teacher of abu Bakr al-Khateeb al- Baghdadi who read/recited the Sahih in front of her she died without being married.
Ibn Shabuwyah محمد بن عمر بن شَبُّويه الشَّبّويُّ، المروزيُّ (died 378 a.H. or later). He was -according a-Dhahabi- among the great Sufi scholars, he met al-Farbari and heard the Sahih 316 a.H.. he was teaching as-Sahih 378 a.H. in Marw. Among his students was Sa'id ibn abi al-'Iyar سعيد بن أبي العيار (could also be a sufi who died 457 a.H. in Ghazna). His status as narrator is rather good however scholars had preservation against him due to his narration of strange books and sources. After the death of ibn Shabuwayh people looked for the narration of al-Kushmihany.
Abu 'Aly ibn as-Sakan (al-Hafidh) (294 a.H. -353 a.H.) الحافظ أبو علي سعيد بن عثمان ابن السكن whom authered a book which was called Sahih ibn as-Sakan. A-Dhahabi claimed that most of his hadith was known in al-Maghrib and rather hard to gather in al-Mashriq. His copy of Sahih al-Bukahri was the first to reach Egypt. Among his narrators of al-Bukhari's as-Sahih are mainly scholars from al-Andalus like 'Abdullah ibn Muhammad ibn Asad al-Qurtubi عبد الله بن محمد بن أسد القُرْطُبِي (310 a.H. -390 a.H.) Abu Ja'afar ibn 'Awn Allah al-Qurtubi أبو جعفر بن عون الله (300 a.H. - 378 a.H.) and the Qadi abu 'Abdullah Muhammad ibn Ahmad ibn Mufarrij أبو عبد الله محمد بن أحمد بن مُفَرِّج (315 a.H. - 380 a.H.).
Abu Zayd Al-Marwazzy محمد بن أحمد بن عبد الله بن محمد المروزي (301 a.H. -371 a.H.). Among his narrators of the Sahih are 'Abdullah ibn Ibraheem al-Asylee who went out seeking knowledge in Mekka around 351 a.H. and introduced it in what is now known by Morocco (died 392 a.H.), the author of al-Hilya abu Nuay'm al-Isbahani (336 a.H. - 430 a.H.) and the author of al-Mustadrak al-Hakim from Nishapur. He himself met al-Farbari 318 a.H. according his own statement. A copy of his manuscript which was compiled around 370 a.H. is the oldest known existing copy of Sahih al-Bukhari.
Abu Ahmad al-Jurjani محمد بن محمد بن يوسف بن مكي، الجرجاني (died 373 a.H.) both ibn Nua'ym and al-Khateeb al-Baghdadi confirmed that he thought as-Sahih, while ibn Nu'aym only confirmed hearing it from him 350 a.H. and said that people declared him as da'if (a-Dhahabi toke this claim and added him in his book on du'afa'), al-Khateeb confirmed that he went to Baghdad to teach it, and that abu Nu'aym told him that he heard a part of it there 357 a.H.. While Muhammad ibn 'Abdal-'Aziz al-Qassar mentioned his journey in Shiraz for the same purpose. other reports say he thought it in several locations such as Egypt (Cairo?), a-Shaam (Damscus?) and Basra. Note that al-Jurjani narrated also from ibn abi Dawod who was declared as very weak by his father.
The grandson of al-Farbari: abu Muhammad Ahmad ibn 'Abdullah ibn Yusuf al-Frabari أبو محمد، أحمد بن عبد الله بن محمد بن يوسف، الفَرَبْريّ (died 371 a.H.) only the historian of Bukhara Ghunjar غُنْجار is noted among his narrators of the Sahih however al-Khateeb al-Baghdadi and ibn 'Asakir named other narrators who heard from him books of his grandfather who had many (copies of) manuscripts of imam al-Bukhari and also about him.
An'Nuaymi also known as-Sarkhasi أحمد بن عبد الله بن نُعيم بن الخليل، أبو حامد النُّعَيميّ، السَّرْخَسي (died 386 a.H.). Among his studnets was abu 'Umar al-Maliyhi al-Harawi أبو عمر المليحي الهَرَويّ who dies 463 a.H. who was the teacher of imam al-Baghawi البَغَويّ the author of Sharh as-Sunnah شرح السنة.
Abu Bakr al-Ishtikhny محمد بن أحمد بن محمد بن مَتَّ أبو بكر السمرقندي الإشتيخني (died 381 a.H. some said 388 a.H.).
Abu 'Aly al-Kushshany أبو علي إسماعيل بن محمد بن أبي نصر محمد بن أحمد بن حاجب بن خمانة الكُشّانيّ الحاجبي الخماني (died 391 a.H.). Most scholars say he was the last to hear from al-Frabari as they met 320 a.H.. Among his narrators was mentioned Ghunjar.
Abu Nasr al-Akhsykati أبو نصر أحمد بن محمد بن أحمد الأَخْسِيكَتي (died 346 a.H.) ibn Hajar mentioned some of his narrators.
Muhammad ibn Khalid al-Farbari محمد بن خالد الفَرَبْريّ, his narration was mentiond by al-Khattabi and ibn Nuqtah.
Abu Luqmaan Yahya ibn 'Ammar al-Khatlany أبي لقمان يحيى بن عمار الختلاني was mentioned by al-Kandahlawi in his introduction of his commentary of Sahih al-Bukhari where he clained that this man lived 143 years.
Copies that were famous in the 6th hijri century
As for the most prominent compilations of Sahih al-Bukhari in the sixth century these were four:
- The narration of abu Dahr al-Harawi who was a student of three students of al-Farbari: al-Mustamli (in the year 374 a.H. till 375 a.H. by both reading/reciting and listening), al-Hammuwi (in 371 a.H.) and al-Kushmihany (in 389 a.H. both by reading and listening.
- The narration of abu Muhammad 'Abdullah ibn Ibraheem al-Aseely it was narrated from al-Farbari and his student al-Jurjani
- The narration of abu al-Waqt أبو الوَقْت عبد الأول بن الشيخ المحدث المعمّر أبي عبد الله عيسى بن شعيب بن السجزي ثم الهَرَويّ، الماليني (died 533 a.H.) which is a narration via a-Daawudi from al-Hammuwi from al-Farbari.
- The narration of Karima al-Marwazziyah كريمة بنت أحمد بن محمد بن حاتم المَرْوَزيّة (died 463 a.H.).
For further information and deeper insight in how the Sahih al-Bukhari we have at hand today was "released" you may refer to my main reference the study of Dr. Jumu'ah Fathi 'AbdalHaleem جمعة فتحي عبد الحليم called كتاب روايات الجامع الصحيح ونسخه «دراسة نظرية تطبيقية» which discusses the variations of the narration of Sahih al-Bukhari and it's copies.
Also note that many commentaries of Sahih al-Bukhari discuss the above topic in their introduction in the commentary of ibn al-Mullaqin ابن الملقن (723 a.H. -804 a.H.) called A-Tawddyh li sharhi al-Jami'(i) as-Sahih التوضيح لشرح الجامع الصحيح (literally: The clarification to explain al-Jami' as-Sahih) there's a chapter addressing your question (see here in Arabic). This book is available in 36 volumes and was compiled between 763 a.H. and 785 a.H.. Ibn Mullaqqin was a teacher of al-hafidh ibn Hajar al-'Asqalani and the historian al-Maqryzi.
See also the article shared by @TheZ.