Like most large empires in the pre-modern period, the Mongols went down to fragmentation and infighting. Even before the death of Ghengis he engaged one of his sons in war (Jochi) and disinherited both him and the brother with whom he feuded (Chagatai). Though Jochi was killed, Chagatai would go on to run his own semi-autonomous kingdom within the wider Mongol Empire referred to widely as the Chagatai Khanate. The Golden Horde was also descended from this conflict, Batu and Orda, the first co-rulers of the Horde (sometimes referred to as the Ulus of Jochi) were his sons.

The children of Ogedai, Ghengis' successor would again split the empire, with the Golden Horde and Ilkhanate becoming effectively autonomous in the same way the Chagatai Khanate was. The succession of every Khan following Ghengis was marked by at least rumblings of discontent and often by outbreaks of violence (you could even argue that only by murdering one of his sons and establishing the other as a client king did Ghengis avoid civil war himself). By the time Kublai - who himself only grasped power after the longest and bloodiest of the Mongol civil wars - established the Yuan Dynasty, the main 'Mongol Empire' no longer held any real authority over the other states. (Chagatai Khanate, Golden Horde, Ilkhanate)

Later rulers described by popular history as Mongol Emperors (eg Timur the Lame/Temurlane) weren't actually in control of the same entity as the one founded by Ghengis. The various rump states would all steadily fall to further infighting and fragmentation but that's spread over a period of more than 300 years. They don't rule the world because the truly successful Mongol Khagans (Ghengis and Ogedai) were not successful due to a well run state or a carefully administered bureaucracy but due to the force of their personality, and when they were dead and gone their successors were unable to maintain the enormous chimaera of an emperor they had been left.