How did the conquering Mongols and Manchus of China end up becoming part of China?

The main change was that the Age of Cavalry ended.

From roughly the popularization of the stirrup to that of gunpowder-using infantry, Cavalry was the dominant military branch. Infantry during this period was essentially only useful as support units for the cavalry (particularly against other non-cavalry units).

A good cavalry force required a lifetime of training, which meant the strongest military forces tended to come from places where people already spent their lifetime in the saddle: the Eurasian steppe.

The steppe dwellers nearest China were Turks, Mongols and Manchus, so they were a constant threat during this period.

Settled farming societies are good at providing large infantry armies, but in an era when infantry was not the decisive arm, this didn't help them a lot. They were reduced to trying to bribe the pasoralists, and failing that, trying to culturally absorb their conquerers.

However, when gunpowder made large infantry units the dominant arm, the situation reversed. Suddenly the relatively underpopulated pastoral regions became power vaccums. The Mongols managed to keep some semblance of a country together during this period, but the Manchurians held more strategic coastal territory, and thus weren't so lucky. They essentially got divied up amongst China, Russia, and Japan, based on those power's relative performace to each other on the battlefield.

Everyone think China is solely Han. But actually China is a combination of hundreds of races other than Han. The Han itself is also combination of few other hundreds small races with their own dialect languages, cultures and believes. Adding Manchu in its dominion is like adding 1 more race into hundreds others. Mongol and Manchu was never fully conquered in the ancient times as Chinese Emperors believed anything beyond the great wall are uncivilized barbarians who barely can be profitted with trades.

When Republics in China were born (I am talking about first nation called China) it only concentrated within Southern area and the Machurian rules the Northern part. After years of civil wars, WW2, Cold war, the Communists emerged as the dominant power. Like other nations around the world, China entered the phase of Expansionism. Invasion through Uighur, Tibet and Manchu land expanded China over than twice of its territory.

But the question is, why China won't take Mongol. The reason is because Mao does not trust Stalin in term of alliance who set his eyes to conquer the East. It will take long route for the Russian army into heart land of China as they have to pass Mongol. If China conquered Mongolia, the Russian army will have rapid access down south at anytime. Manchu on the other hand is safer to be conquered as it open access for China to cross to Korean peninsula.

Now China has grown to most largest military in quantity and second largest in quality. China however interests more in Southern riches than barren North. We won't see China invading mongol anytime soon, but they are ready to take islands from South East asia nations who are riches of Gas, Oil, and food.

The Mongols and Manchus elected to "join" China upon conquering it, because the latter was more advanced and civilized. So upon conquering China, they just took over the Chinese cities, palaces, country, for themselves, and installed themselves as the ruling class.

After the death of Genghis Khan, the "father" of the Mongol Empire, it divided into four parts; the Golden Horde (modern Russia), the Ilkhanate (the Middle East), Chagatai (the modern Kazazh- and other -stans), and China-Mongolia, which fell to Genghis' grandson Mongke, the older brother of Kublai Khan. Kublai was sent to China to "run the family business" (conquer the rest of China), so he set up a "headquarters" where Beijing now is. When Mongke died, Kublai inherited the easternmost Mongol kingdom, which Kublai preferred to rule from China. His younger brother Ariq Boke, occupied Mongolia proper and started a civil war, which Kublai won, leading to the virtual destruction of Karakorum, the Mongol capital So Kublai only had (the modern) "Beijing" (under a Mongol name) as his capital. But the civil war destroyed whatever (weak) claims Kublai had to the other three kingdoms, so he was left with China (and a ravaged Mongolia). Kublai's descendants ruled as the Yuan Dynasty until 1368 (rise of the Ming dynasty).

The Manchus were in a similar situation. Although they had a few (relatively small) cities, they were mostly a bunch of nomadic (Jurchen) tribes looking for a place to settle. As was the case with the Mongols, Beijing and other Chinese cities were larger and richer than the ones they knew, meaning that they were glad to conquer, occupy, and rule them.

Put another way, the Mongols and Manchus elected to "move into" (rather than "take home") China. In so doing, they "became part of" China.