For some reason I thought I wrote a post on this, but then it vanished. That's what I get for doing it at 2:00 AM.
During the 1800's Mongolia was part of the Qing Dynasty which, of course, means that it was nominally controlled. Officially, the Emperor held the title of Khan of Mongolia, and Mongolians were considered lesser people (forbidden to marry Chinese, etc.). For much of the period, the area was a backwater, ignored so the the vast expanse of Outer Mongolia could serve as a buffer to Tsarist Russia. In the latter half of the century, deliberate efforts were made to settle ethnic Chinese in Inner Mongolia to bolster the frontier.
Russian adventurism was never heavy in Mongolia, mostly focused on Manchuria or Kazakastan. Toward the end of the century, there were more moves on western Mongolia, which in turn alarmed the Chinese.
Around 1900, Mongolian nationalists made overtures to the Russians, as natural enemies of the Chinese. Nothing came of it, but region declared independence in 1911 under Bogd Khan (with Russian and Japanese support). That lasted for several years until the a Chinese warlord removed him, but then in 1921 he was briefly restored by Baron Ungern-Sternberg (the Bloody Baron). But that takes us into the whole Back of Beyond time.
So, during this time you've got a mix of traditional Mongol nomads, Qing troops (pre-Boxer armies and probably some reformed post-Boxer units), and local troops. During the independence period you can add Russian and Japanese advisors providing weapons and training, a standing Mongol "army", Chinese warlords, and eventually White Russians and Bolsheviks. Ethnic riots, decadent Mongolian nobles, oppressive tax collectors, troop mutinies, independence movements, Chinese traders, Russian (and some Japanese) agents, exiled anti-imperial Han, bandits, and Western adventurers are all good fodder for a game in this period.