For example, are there any records of prophets being sent by Allah to China, Europe, India, or Africa?
Clear answers from the qur'an
According the qur'an certainly "yes" read for example:
[We sent] messengers as bringers of good tidings and warners so that mankind will have no argument against Allah after the messengers. And ever is Allah Exalted in Might and Wise. (4:165)
... You are only a warner, and for every people is a guide. (13:7)
And We certainly sent into every nation a messenger, [saying], "Worship Allah and avoid Taghut." And among them were those whom Allah guided, and among them were those upon whom error was [deservedly] decreed. So proceed through the earth and observe how was the end of the deniers. (16:36)
Indeed, We have sent you with the truth as a bringer of good tidings and a warner. And there was no nation but that there had passed within it a warner. (35:24)
All the above verses say that Allah has sent to every nation warners (prophets or messengers).
Not all messengers and prophets have been mentioned
One may ask why all the known prophets () are only sent to middle east, here some answers:
One within the qur'an, Allah the almighty said:
And [We sent] messengers about whom We have related [their stories] to you before and messengers about whom We have not related to you. And Allah spoke to Moses with [direct] speech. (4:164)
And We have already sent messengers before you. Among them are those [whose stories] We have related to you, and among them are those [whose stories] We have not related to you. And it was not for any messenger to bring a sign [or verse] except by permission of Allah . So when the command of Allah comes, it will be concluded in truth, and the falsifiers will thereupon lose [all]. (40:78)
So not all messengers () and prophets () are quoted by name in the qur'an.
Many ahadith quoted high amounts of messenger of hundred of thousands.
Why mainly only prophets from middle east have been mentioned
And Allah further told the prophet () their stories for some reasons among them:
And certainly were messengers denied before you, but they were patient over [the effects of] denial, and they were harmed until Our victory came to them. And none can alter the words of Allah . And there has certainly come to you some information about the [previous] messengers. (6:34)
And each [story] We relate to you from the news of the messengers is that by which We make firm your heart. And there has come to you, in this, the truth and an instruction and a reminder for the believers. (11:120)
To help Muhammad() to support his task by quoting people (prophets and messengers) he knew or might have heard from and to remind the believers too.
The first verse above also shows that disbelievers deny messengers and prophets, so they could hide about having received a message or revelation via prophets, and don't mention them!
Sheikh ibn 'Ashur for example said:
Allah did not tell the Prophet (blessings and peace of Allah be upon him) the names of many of the Messengers, and only told him the names of a few, because those who are mentioned are the greatest of the Messengers and Prophets and greater lessons may be learned from their stories. End quote.
At-Tahreer wa’t-Tanweer, 6/35 (Source: islamqa #138770)
Further in the same fatwa as above you may find the explanations:
It is well known that most of the civilisations that were established throughout history were in this area, which is known as the Mediterranean basin, and nearby, in Syria, Egypt, Iraq and Arabia. Hence the most populated areas were mostly in these lands. So it is very appropriate that most of the Messengers should have been sent to the people of these lands.
Because those Messengers had lived in the region inhabited by the Arabs and thereabouts, their stories were known to the Arabs and to the People of the Book who lived in those regions. This was more effective in establishing proof against them and the lessons of what befell those people has a greater impact.
And that there is no nation in which there is no warner; The life of my nation is Nazir and Hashdar Dhandah, i.e., the presence of Dard.” Nez: “We have sent a messenger to every nation to worship God and avoid the tyrants;” We sent to every nation a messenger (to tell the people) that you should worship God and turn away from idols. ”(1) The divine prophets were trained human beings who had been commissioned by God to convey the commands of God and to guide human beings. The tradition of God was to motivate a person to lead them among each group and ethnic group. As he says in the Holy Qur'an: "We have raised a messenger in every nation [".  This theme has been mentioned in other verses of the Holy Quran.  So it is certain that prophets have been sent to guide people everywhere in the world, 
Therefore, from the beginning of creation, wherever human beings lived, God sent many prophets to express their duties to them, but the names and stories of all of them are not described in detail in the scriptures, including the Qur'an. .  .Nahl, 36. . Fatir, 24; اسراء, 15; Taha, 134. . In addition, divine wisdom and justice require the guidance of individuals in every age and generation. . Existence of Prophets in Non-Asia and the Middle East, 2503; "Names of the Prophets", 32102; "The burial place of some prophets", 12648.