In respect to the Mongols - Kublai Khan established the Yuan dynasty, but the empire that Genghis Khan established was split into four different khanates - Khanate of the Golden Horde, Chagatai Khanate, Ilkhanate, and Khanate of the Great Khan (or the Yuan dynasty of China). While to a certain extent these different groups aided each other, they were effectively split due to various internal struggles/conflicts (e.g. who was the rightful heir) as well as external influences (e.g. spread of Islamic religion).

A large reason of why the Mongol Empire was so successful in dominating large regions of Asia was due to the fact that Genghis managed to unite the various Mongol tribes of the Eurasian Steppe. Were it not for this unification under one person, they would most likely have never been able to establish themselves as an empire.

Similarly, the Mongol Empire never reformed to the same state that they were under Genghis' rule - after the establishment of the Ming dynasty in China that forced the ruling Yuan Emperors out, they retreated to Mongolian homeland where they continued rule as Post-Imperial Mongolia. However, we do know of one person, Timur (or Tamerlane) who established his own dynasty in Central Asia reconquering most of the regions once controlled by the Chagatai Khanate and the Ilkhanate, but I'm not well enough informed on that to provide a proper answer.

The Umayyad Caliphate, while representing the apex of 'Islamic empires' to people who only pay attention to land mass size, kept power closely concentrated in the Arab elites. It was not until the Abbasid revolution that power would expand to all Muslims in the caliphate, not just those who were Arab by blood. This was a cooperation between Arab Sunnis who did not think racism should be a part of their government, and also Shi'a who considered the Umayyads 'evil' because of their role in massacring their Imam Hussein at Karbala on Ashura.

However, once the Abbasid caliph - the relatives of Abbas, Muhammad's uncle - came into power, they purged those who had helped them come to that power, including Abu Muslim al-Khorasani, 'The Son of a Muslim from Khorasan', the leader of a great number of non-Arab Muslims whose armies had helped defeat the Umayyad Caliphate. Anti-Shi'aism was also a mainstay of Abbasid rule, although depending on the Caliph, some were more anti-Shia, or pro-Shia than others. The capital was moved from Umayyad Damascus to Abbasid Baghdad. The 'House of Wisdom' from which Islamic learning would become so famed was based in Abbasid times in Baghdad, not Umayyad times in Damascus. Previously the lands around Baghdad were the capital of the Sassanian Empire in Persian times, and for that reasons many Persians would be found in the administration of the Abbasid Caliphate, but not the Umayyad Caliphate who were strictly feudalistic.

Epilogue: The Umayyad Caliph didn't quite disappear, but survivors reappeared in Spain as the 'Umayyad Caliphate of Cordoba', lasting many years until it divided into Taifas that were one by one conquered by the Catholic Monarchs of Spain, and occasionally conquered by Moroccan Caliphates and Sultanates. By the end of the Reconquista the Kingdom of Granada was all that remained, a Muslim feudal dynasty that allied with the Catholic Monarchs against the Moroccans. These people would be expelled from Spain after being forcibly converted yet regarded as 'crypto-Muslims' by the Catholic Monarchs. This suspicions also extended to the Jews who lived in the Kingdom of Granada at the time, and they were expelled as well. The Cordoban diaspora spread to African and Asian states, but also interestingly enough many of these people entered France, which did not have that same sort of religious discrimination as part of their rule. (They had a different kind, but it is not comparable to zealots of the 'Reconquista')

Epilogue to Abbasids: The Abbasids were vividly annihilated by the invading Mongol hordes, but survivors continued to claim the Abbasid Caliphate under the 'protection' Ayyubid Sultanate of Egypt. The Mamelukes displaced the Ayyubids as 'protectors of the Caliph'. This lasted until the Turkish Sultan defeated the Mamelukes and forced the Abbasid Caliph to turn his symbols of power over to the Turkish sovereign. To this degree, the Ottoman Empire was an attempt of revitalization or continuation of the original Caliphate. In 1924, when the Ottoman Caliph is unseated by the new Turkish parliament, the Hashemite ruler of Hejaz declared his own 'Sharifah Caliphate' using his control of Hejaz as his mark of rule, until he was swiftly conquered by Ibn Saud. Although he named his Hashemite successor as the next Caliph, they chose not to adopt the title themselves, and that is why today there is no continuous claimant to the Caliphate. Although, looking at the history, you can see it became a feudal 'prize' a long time before.

During WWI, native Indians and Pakistanis both tried to inspire the British government to 'protect the caliphate' in the form of the Khalifat movement. This was suppressed as an insurgency, and today's 'protectors' or 're-establishers' of Caliphate can trace their descendant largely to the chaos that erupted at the close of WWI. Contemporary jihadists believe themselves certainly to be 'restorers of the caliphate', whether Umayyad, Abbasid, Fatimid is up to them and their specific theologies. Racists may favor the Umayyads, pragmatists the Abbasids, and Shi'a the Fatimids. In fact, the Fatimid line is preserved in the Ismaili sect, who see the leader of their movement as being a descendant of the Fatimids and successor of an uninterrupteed line of caliphs going all the way back to Ali (making them Shi'a Ali - Party of Ali), and then a step further to Muhammad. Other 'alternate caliphates' exists, ranging from the universally peaceful Ahmadis (who are regarded as non-Muslims by excommunicators) to Daesh (who are regarded as Khawarij, a fanatical sect who killed Ali, leading to the downfall of the first Rashidun Caliphate or 'Rightly Guided' Caliphate).

Salafists do not claim to be Caliphs, but claim to follow the example of the Rashidun Caliphs by casting away the hundreds of years of governments that succeeded the Rashiduns in an attempt to get 'back to the original way'. To that degree, they may not claim 'caliphate' but they are certainly trying to be 'successors.'