The caliphate is among Quraysh

It is narrated from Urwah bin al-Zubayr (may Allah have mercy on him) that Musaylimah (the Liar) sent a letter to the Messenger of Allah (peace and blessings be upon him) whose content was: "From Musaylimah bin Habib to Muhammad, the Messenger of Allah (peace and blessings be upon him). Peace be upon you. To proceed: half of the land belongs to Quraysh and half of the land is ours, but Quraysh are a people who transgress and commit aggression." Those two messengers testified that Muhammad (peace and blessings be upon him) is the Messenger of Allah, and they further said: "Musaylimah does not deny this fact, except that he too has been made a partner with you in the matter of prophethood, and prophethood has been given to him along with your prophethood."
The chain of this narration contains irsal (a break).

Zayd bin Abi Attab relates that Sayyiduna Mu'awiyah (may Allah be pleased with him) stood on the pulpit and said: I heard the Messenger of Allah (peace and blessings be upon him) say: "People follow Quraysh in this matter (of rule and caliphate). Those of them who were best in the time of Jahiliyyah are also the best after embracing Islam, provided they acquire fiqh (understanding of the religion). And were it not that I fear Quraysh would become proud and arrogant, I would tell them all that is with Allah Almighty for them (of reward and high rank)."
I (Ibn Hajar) say: Imam Ahmad bin Hanbal (may Allah have mercy on him) also narrated with this same chain: "The best women who ride camels are the women of Quraysh" (the complete hadith).

It is narrated from the grandfather of Kathir bin Abdullah (Sayyiduna Amr bin Awf al-Muzani, may Allah be pleased with him) that he said: I was present with the Prophet (peace and blessings be upon him), and he (peace and blessings be upon him) said: "O assembly of Quraysh! After me you are the guardians and inheritors of this affair (of caliphate and rule); 'And do not die except as Muslims' [Surah Al Imran: 102] and 'Hold fast, all together, to the rope of Allah' [Surah Al Imran: 103] (to the end of the verse); and take care of my right regarding the Ansar, their sons and their grandsons (that is, honour my right by treating them well)."
Kathir (the narrator) is da'if (weak).

With the same previous chain it is narrated: I was sitting in the blessed gathering of the Prophet (peace and blessings be upon him) when he (peace and blessings be upon him) said: "The ally (halif) of a people is counted among them, and the sister's son of a people is also (counted) among them."
Imam al-Bukhari narrated this hadith in al-Adab al-Mufrad, and this narration is sahih (authentic).

It is narrated from Sayyiduna Ali (may Allah be pleased with him) that one day the Messenger of Allah (peace and blessings be upon him), addressing the people in a sermon, said: "Beware! The rulers will indeed be from Quraysh; listen! The leaders will be from Quraysh, as long as they hold to three qualities: when they judge, they do justice; when they make a covenant, they fulfil it; and when mercy is asked of them, they show mercy. So whoever of them (the rulers) does not do so, upon him will be the curse of Allah Almighty, of the angels and of all mankind."
Imam Abu Ya'la (may Allah have mercy on him) narrated this hadith in his book Musnad Abi Ya'la al-Mawsili.

It is narrated from Abdullah bin Urwah that a year of severe drought brought Nabighah bin Ja'dah to us while we were with Sayyiduna Ibn al-Zubayr (may Allah be pleased with him) in Makkah. After leading the people in the dawn prayer in al-Masjid al-Haram, he stood up and recited these verses: "By becoming our governor you have revived for us the memory of Sayyiduna al-Siddiq, Sayyiduna Uthman and Sayyiduna al-Faruq (may Allah be pleased with them), and the destitute have found relief. Abu Layla (Nabighah) has come to you cleaving the darkness of the nights, a strong traveller crossing the deserts, so that you may set right his condition, which the turns of night and day and a harsh time have scattered." Sayyiduna Ibn al-Zubayr (may Allah be pleased with him) said to him: "O Abu Layla! Enough, for this poetry of yours is in our view the least of your means (of obtaining wealth). As for our pure personal wealth, it is reserved for the family of al-Zubayr; and as for our surplus wealth, the needs of Banu Asad keep us from turning to you. However, you have two rights in the treasury: one right because of the honour of having seen the Messenger of Allah (peace and blessings be upon him), and the other right because of your partnership with the Muslims at large." Then he ordered that his mounts be loaded with grain and dates. Abu Layla began hurriedly eating the dates and grain, at which Sayyiduna Ibn al-Zubayr (may Allah be pleased with him) said: "O Abu Layla! Great hardship and severe hunger are showing on you." When he had satisfied his need, he said: I bear witness that I heard the Messenger of Allah (peace and blessings be upon him) say: "As long as Quraysh rule and act with justice, show mercy when mercy is asked of them, speak the truth when they speak, and fulfil when they promise good, then I and all the prophets (peace be upon them) will be forerunners at the Pond (al-Kawthar) for those al-qasifun." The narrator said that al-qasifun are those who crowd at the Pond to drink all at once.
Ibn Abi Umar said that here the word al-mal (wealth) means camels.