Islam will be dominant till the time of the twelve Caliphs:

Prophet Muhammad (SallAllah-u-Alaihi-wa-Sallam) said: This Khilafah will not end until there have been twelve Khulafa among them, all of whom will be from the Quraish.
[Sahih Muslim, Book 020, Number 4477]

Prophet (SallAllah-u-Alaihi-wa-Sallam) said: Islam will continue to be dominant until there have been twelve Khulafa, all of them will be from the Quraish.
[Sahih Muslim, Book 020, Number 4480]

It is clear from the above ahaadith that Islam will be dominant in the era of the twelve Khulafa, the Khilafah will not end until the twelve Khulafa at least and all will be from the Quraish. This proves that the first twelve Khulafa after the Prophet (SallAllah-u-Alaihi-wa-Sallam) kept the Deen dominant as all them were from the Quraish tribe, if they were any individual mistakes they cannot be taken as proof against them.

Rule in Quraish with a Condition:

Narrated Muhammad Ibn Jubair Ibn Mut’im (RadiAllahu Tala Anho): That while he was included in a delegation of Quraish staying with Muawiya (RadiAllahu Tala Anho), Muawiya heard that ‘Abdullah bin ‘Amr (RadiAllahu Tala Anho) had said that there would be a king from Qahtan tribe, whereupon he became very angry. He stood up, and after glorifying and praising Allah as He deserves, said, ‘To proceed, I have come to know that some of you men are narrating things which are neither in Allah’s Book, nor has been mentioned by Allah’s prophet (SallAllah-u-Alaihi-wa-Sallam). Such people are the ignorant among you. Beware of such vain desires that mislead those who have them. I have heard Allah’s prophet (SallAllah-u-Alaihi-wa-Sallam) saying: This matter (rule) will remain with the Quraish, and none will rebel against them, but Allah will throw him down on his face, as long as the Quraish uphold the Deen.
[Sahih Bukhari, Volume 9, Book 89, Number 253]

Muawiya (RadiAllahu Tala Anho) misunderstood Abdullah bin Aamr (RadiAllahu Tala Anho) as he was only narrating a hadith regarding the king from Qahtan tribe, this particular hadith can be reviewed in Sahih Bukhari, Kitab ul Fitan.

Muawiya (RadiAllahu Tala Anho) thought it to be wrong that the rule might get out of the hands of Quraish, however if he had reviewed the hadith he narrated himself that the rule in Quraish is with a condition then this misunderstanding might not have taken place.

Abu Bakr (RadiAllahu Tala Anho) said: This rule will remain in Quraish till they keep obeying Allah and stay on His ruling.
[Fath ul Bari 13/116, Sanad: Sahih]

Prophet Muhammad (SallAllah-u-Alaihi-wa-Sallam) said: This Imara was in the Hameer meaning non Quraish then Allah snatched it from them and gave it to the Quraish and soon it will return to the Non Quraish.
[Fath ul Bari 13/117, Sanad: Jayyad]

The condition once again highlighted is that they uphold the Deen. When the Quraish were not able to uphold the Deen then Khilafah went out of their hands as according to the above hadith.

The Quraish are meant to be Rulers:

Messenger of Allah (SallAllah-u-Alaihi-wa-Sallam) said: People are the followers of Quraish in good as well as evil (pre-Islamic times).
[Sahih Muslim, Book 020, Number 4475]

Allah’s prophet (SallAllah-u-Alaihi-wa-Sallam) said: This matter (rule) will remain with the Quraish even if only two from them existed.
[Sahih Bukhari, Volume 9, Book 89, Number 254]

Two Conditions for Precedence:

Prophet (SallAllah-u-Alaihi-wa-Sallam) said: The tribe of Quraish has precedence over the people in this connection (i.e. the right of ruling). The Muslims follow the Muslims amongst them, and the infidels follow the infidels amongst them. People are of different natures: The best amongst them in the pre-Islamic period are the best in Islam provided they comprehend the religious knowledge…
[Sahih Bukhari: Volume 4, Book 56, Number 700]

Muawiya (RadiAllahu Tala Anho) narrates that the Prophet (SallAllah-u-Alaihi-wa-Sallam) said: People will remain followers of Quraish in this Imara, those who were respectable in pre Islamic times will remain respectable after accepting Islam till they have understanding (of Deen)…‘
[Fath ur Rabbani part 23, pg 9, Sanad: Jayyad]

Prophet (SallAllah-u-Alaihi-wa-Sallam) said: Your father is one. Beware! No Arab has precedence over an Ajmi, no Ajmi has precedence over an Arab, No white has precedence over a Black and no Black has precedence over a white, there is no precedence but on the basis of Taqwa (piety).
[Musnad Ahmad, Buloogh ul Amani part 12, pg 226, Sanad: Sahih]

From the above ahaadith it is clear that Quraish cannot be given precedence over others, if others have knowledge of Deen and are more righteous.

A Slave can also be an Ameer:

Allah’s prophet (SallAllah-u-Alaihi-wa-Sallam) said: You should listen to and obey your ruler even if he was an Ethiopian (black) slave whose head looks like a raisin.
[Sahih Bukhari, Volume 9, Book 89, Number 256]

Muawiya (RadiAllahu Tala Anho) addressed the people and said: Oh people you have given baiyah to me happily and willingly, however if you had given baiyah to an Ethiopian slave maimed and disabled, I would have came and given baiyah to such a slave with you…
[Ibn e Abi Shaiba 11/94, Sanad: Sahih]

When an Ameer (leader) has been appointed we cannot go about disobeying him or try to replace him as when appointed he is to be obeyed in matters of administration.

Muawiya (RadiAllahu Tala Anho) who narrated the hadith regarding Quraish being rulers, himself accepts when baiyah is already given then such an Ameer should be obeyed whether from any background.

Conclusion:

So it is clear that the rule should remain in Quraish if they are worthy of it, whoever tries to take it away from them then Allah will repulse such people. However when an Ameer is appointed whether of any background then listen and obey such an Ameer.

Alhamdulillah the previous Ameer of Jamaat-ul-Muslimeen ‘Masood Ahmad’ was from the Quraish and so is the present Ameer ‘Mohammad Ishtiaq’, but they weren’t made Ameer because they belonged to Quraish, but because of their knowledge of Deen and Taqwa, Allah is the best judge.

Unfortunately as there have been more then 1400 years since Prophet (SallAllah-u-Alaihi-wa-Sallam) left us so there isn’t a possible way for verifying anyone’s claim of being Quraishi until such a person is a relative of Imam Mahdi who is yet to be seen.