Some sunni believe (including myself used to believe) mahdi will be final caliph from quraysh. Based on a hadith
Sahih Muslim 1821 e It has been narrated on the authority of Jabir b. Samura that the Prophet (ﷺ) said: This order will continue to be dominant until there have been twelve Caliphs. The narrator says: Then he said something which I could not understand, and I said to my father: What did he say? My father told me that he said that all of them (Caliphs) would be from the Quraish.
It has been narrated on the authority of 'Abdullah that the Messenger of Allah (ﷺ) said: The Caliphate will remain among the Quraish even if only two persons are left (on the earth). — Sahih Muslim 33/4
'Asim narrated from Zirr, from 'Abdullah, from the Prophet(s.a.w) who said: "A man is coming from the people of my family whose name agrees with my name." 'Asim said: "Abu Salih narrated to us from Abu Hurairah, who said: 'If there did not remain in the world but one day, then Allah would extend that day until he comes.'" Jami` at-Tirmidhi 2231
Can't male offspring of mahdi be caliphs after mahdis death and death of eesa bin maryam?
Eesa will rule for 40 years as per some hadith commentators.
من وجه آخر عن أبي هريرة أقرئوه من رسول الله السلام وعند أحمد من حديث عائشة ويمكث عيسى في الأرض أربعين سنة وللطبراني من حديث عبد الله بن مغفل ينزل عيسى ابن مريم مصدقا بمحمد على ملته And it was narrated by Ahmad from another route on the authority of Abu Huraira, who had read it (or been informed) (from others) from the Messenger of Allah, peace be upon him, and according to Ahmad from the hadith of 'Aisha, that 'Isa will remain on earth for forty years, and by al-Tabarani from the hadith of 'Abdullah bin Mughaffal, 'Isa son of Mary descends, confirming Muhammad on his religion (millah)