Short answer:

Yes, there could be, but "how much" begs the question: "who are you asking about, when?"

Discussion:

Mobility could be substantial, but it varies in place and time -- there's nothing useful that can be said about "Ancient China" as a generality -- that's a bit like asking about "Europe"; this is millions upon millions of people, through thousands of years, natural disasters, invasions, revolts and so on. Sometimes ruled by Han Chinese, sometimes by Khitans, sometimes by Mongols, sometimes by Jurchen.

Anything that might be true at some place and time might not be true at another. There's also a vagueness to the term "commoner" and "aristocrat" -- Chinese society had many more gradations than that. China did become more bureaucratic over time, albeit not in any linear progression, just a tendency to greater centralization, one which which fell apart on many occasions. China's great epic, The Romance of the Three Kingdoms, begins with the words

"The world under heaven, after a long period of division, tends to unite; after a long period of union, tends to divide."

. . . and that's a parsimonious summary of the cycles of centralization and fission in Chinese authority.

China has all sorts of documentation that's useful to historians, and quite recently some have begun assembling data from funerary stele to assemble the story of the rise to prominence of a few families, so, for example, considerIiyama, T. (2017). STELES AND STATUS: EVIDENCE FOR THE EMERGENCE OF A NEW ELITE IN YUAN NORTH CHINA. Journal of Chinese History, 1(1), 3-26. doi:10.1017/jch.2016.1

. . . where Professor Iiyama looks at the way that a Chinese family of minor officials navigated the patronage networks of their new Mongol rulers. This is a careful look at a small number of people in one place, at one time.

and similarly Sarah Schneewind has just published "Shrines to Living Men in the Ming Political Cosmos" (Harvard University Press 2018)

-- where she looks as "pre-mortem" monuments, that is monuments that are erected to then living men, as a marker of rising social status during the Ming dynasty. Again, what Schneewind finds here is specific to this place and time, cannot be held to be true of Chinese history generally.

These works all are in a vein that (I think) was inaugurated with Robert Hymes' seminal work:

Hymes, Robert P., Statesmen and Gentlemen: The Elites of Fu-chou, Chiang-hsi, in Northern and Southern Sung (Cambridge: Cambridge University Press, 1986)

. . . which details how government functionaries risen from a meritocratic system cemented their, and their family's status. The socially mobile, having risen, often pull up the ladder that they climbed, in China no less than elsewhere.

Speaking of meritocracy, no discussion of elite mobility could miss China's system of imperial examinations which has origins perhaps 2000 years old, but really takes hold during the Song Dynasty; its then that it becomes the systematic means of filling governmental positions.

See:

Miyazaki, Ichisada (1976), China's Examination Hell: The Civil Service Examinations of Imperial China, tr Conrad Schirokauer, Yale University Press, 1981

Similarly, at some times there was an extraordinary bargain possible for a literate and ambitious individual who was willing to part with something precious-- eunuchs had tremendous power at times, and people did voluntarily give up their "treasures" for a chance to rise in power, difficult as this may be to believe.

Note here that the mobility isn't between "commoners" and "aristocrats" -- it's a movement between minor elites and power in government; to pass the imperial examinations required a remarkably deep literacy (far more characters than anyone reads today) and it would be beyond "common people" to afford the education necessary to prepare for such examinations . . . if by that you mean peasantry.