Sorry I'm so late to reply. An eclectic mixture of debating, rowing and work has kept me very busy recently.

The Mongols were fundamentally very adoptive. When they rocked up and conquered somewhere they were usually pretty content to leave the basic bureaucratic structure as was. Their main addition was appointing someone in an overseer/governor role to keep an eye on things. These guys were known as shahna, basqaqs or darugachi depending on where they were (although sometimes the terms mean slightly different things this makes research a bit of a nightmare). So who were these basqaqs? In the early stages when the Mongols had basically no experience of governing sedentary populations they made heavy use of educated Uighurs and Khitan. The former was a Turkic tribe with some bureaucratic tradition. The latter group was the remnants of an old Chinese dynasty that had nomadic origins, some of them were still present but a lot had fled to Cenrtral Asia at the end of their dynasty and established the Kara-Khitai kingdom. However as time goes on we see other groups and a fair few Mongols assume powerful positions as well.

As you might have noted the government structure above allows for a considerable degree of local autonomy, not just for the cities, but also for the basqaqs. As the area that would become the Ilkhanate (Iran, Azerbijan, as well as bits of Iraq,Turkmenistan, Afghanistan and Pakistan) was pretty far from the centre of the empire it was governed in a particularly loose way at first. This changes when the fourth Great Khan, Mongke Khan, brings it under closer control. This fits in with a general trend of reform under him but was also apparently triggered by the particularly egregious corruption and exploitation of the local Mongol governor, a side effect of the loose rule was common corruption. The area is brought under closer control by Hulagau Khan (Mongke's brother) who reduced the general levels of exploitation and corruption by Mongol governors but did leave local elites with a fair degree of autonomy. Roughly 40 years later, in the reign of Ghazan Khan, we see further centralisation. The moves include introducing uniform weights and measures, reforming tax collection and cleaning up/rationalising the bureaucracy.

Well, hopefully /u/rakony or the like will show up to talk about the other mongol areas, but I can talk about China.

When the Mongols established the Yuan dynasty, it was after the defeat of the Jin dynasty in the North and the Southern Song. The Jin were comprised of the Jurchen (a non-ethnic Chinese people from the North), and when the Jin had captured Northern China, there was a great migration of the ethnic Chinese towards the South. This meant that the North-South proportion of ethnic Chinese went to something like 20%-80%, with the Southern Song encompassing most of the "true Chinese" population. The Mongols first conquered the Jin, and then after Song efforts to recapture Northern parts of China, ultimately conquered the Song as well.

Because of the vast rural agricultural nature of China and the very large population, it was a nightmare for the Khanate to actually rule over the area. The population itself was divided into four major groups: the Mongols, the "Central Asian Auxiliaries" (Naimans, Uighurs, Turkish peoples, and Tanguts), North Chinese, and South Chinese. The Mongols themselves remained the ruling class, while the Auxiliaries took over the roles of government workers, soldiers, and merchants. Additionally, since the Mongolians were often not literate in Chinese, much translation work was taken up by the Uighurs in bilingual government offices. So, while the government structure and reforms were modeled after Chinese dynasties, there was very little Chinese involvement in the government (it was more of a martial rule, as Kublai Khan established the emperor as more of a monarch). Another problem is that the Southern Chinese especially were strictly forbidden from engaging with many aspects of society under Mongolian rule. They were forbidden to marry Mongolians, carry arms, and at certain points forbidden from learning Mongolian or any other foreign language.