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.