Many. The Chinese frequently tried to dominate the Mongolian steppes to permanently end all nomadic threats. Obviously, that didn't always work. And in the case of Genghis Khan, it backfired pretty severely. Some examples of Chinese soldiers marching into the heart of Mongolia include the Han-Xiongnu War, the Tang-Gokturk War, the Tang-Uyghur War, and numerous Ming-Mongol Wars. This does not include the Chinese-nomad wars in which the Chinese didn't set foot on the steppes

The most significant takeover would be the soon-to-be-sinicized Manchurians of the Qing Dynasty conquering the Mongols in the early 1600s. Mongolia would remain part of the Chinese Empire until 1911, when it became briefly independent. It would later fight off another serious Chinese invasion in 1921 with Soviet help. In fact, it became the first Soviet satellite and was therefore treated rather generously

This alliance would result in a combined Soviet-Mongolian force crushing the Japanese thoroughly during WWII at the Battle of Khalkin Gol. It was so devastating that the "Strike North" faction (fight China and the Soviet Union) would give in to the "Strike South" faction (fight China and the United States), leading to the Pacific War. Mongol armored units apparently joined the push into Berlin in 1945

Mongolia was ruled by foreign powers, but the takeovers were slow and subtle, not all at once.

The Manchus subsumed Mongolia into their Empire. I read that the Kangxi Emperor was able to get the Mongol loyalty because he fought Galdan and the Dzungar Khanate. This is related in EMPIRE OF THE STEPPES.

The Kangxi Emperor got the personal allegiance of the four Khalka Khans who he saved from Dzungar domination. Thus the Manchus took over Outer Mongolia.

In 1912 the Manchus fell. Mongolia's rulers deemed themselves released from allegiance to the Manchus. Mongolia slowly fell into a Russian, later Soviet, orbit.