Steppe Empires and their relation with Ch

Manchus never reached to Buryat Mongol. Buryat Mongol came under Russia at that time.

Aside from military campaigns, is there any instance where a people migrated west to east along the steppe?

Ive also heard about western steppes being more fertile.It doenst explain why the eastern steppe armies seemed to have higher populations than the western ones. 

completely agree with this view. many Steppe empires will also have realized this and looked how to get rid of thsi dependence or symbiotic relationship, so they attempted to conquer (at least partially) China, but the obvious failure of those conquests was the loss of Nomad tradition above all, including corruption through civilization, the loss of small but strong and disciplined nomad horse armies as opposed to huge Chinese peasant infantry militias. as a result, those sinified nomad empires were unable to retain control of the steppe and were unable to protect themselves from new emerging powers from the steppe.

In a 1981 edition of the Journal of Asian Studies, Barfield made the argument that most steppe empires thrived on the relation with China, which "funded" the nomads with trade and gave the nomads a political agenda to keep themselves united:

• "The imperial overnment (of the Xiong Nu) was at its most powerful and highly centralized when it conducted foreign affairs and organized military campaigns. It was least powerful and most decentralized at the local level where it ceded political power to the indigenous tribal leaders who were responsible for handling most domestic problems."

Of course, the most convenient "military campaign" would be south on the Chinese border. He also argued:

• "The imperial level of government was financed by drawing on resources from outside the steppe, not by taxing the nomadic animal breeders from within the empire."

i.e. Drawing resources through trade and raiding. Generalizing the argument, he wrote:

• "Nomadic conquerors who created empires along the lines of the Xiong nu and used them to exploit China produced powerful states that survived their founders. Those nomadic conquerors who failed to see that a nomadic state required the support of resources from outside the steppe had their empires collapsed or suffer a power vacuum at their deaths."
  

So essentially, the importance of China to the steppes empires, as argued, was that it provided the nomads with resources. For this reason, Barfield argued that nomads did not perfer to entirely conquer China, but maintain an aggressive relation in order to exact the resources. A long, drawn-out attempt of conquest would be too costly, and on the other extreme, an isolationist policy would not generate the economic activity to "fund" the steppe empire.

Since no nomadic power completely conquered China until the Mongols, this observation appears to hold true for at least 1000 years of history before the 13th century.

The centralized power of the nomadic state was derived from achieving the goals described above. An example, provided in the article is that:

• The notable successes founded along Xiong nu lines were those of the Turks and Uighurs who dominated the steppe from the middle of the sixth century... Both successfully forced China to fund a complex state by extorting subsidies and controlling a vast quantity of silk.
  
  In 840, the Kirghiz successfully destroyed the Uighur state and occupied its heartland, but they proved incapable of recreating the complex relationship that the Uighurs had established with Tang China, which funded the Uighur state. AS a result, no Kirghiz state ever came into existence...nor attain any power over the rest of the steppes.