Russia was an immense empire, spanning from the Baltic to the Pacific, and within their borders, dozens upon dozens of nationalities lived. The Russian Army was, likewise, a multi-ethnic force drawing levies from the various corners of the Empire. The cavalry forces of the Imperial Russian Army included within its ranks several irregular forces drawn from the Eurasian steppes and Central Asia. Most famous, by far, would be the Cossack riders, but also within those ranks would be so called "native" or "tribal" units raised from Tatars, Bashkir, and Kalmyk, to name a few, all of whom can often have features generally considered "Asian" (these groups would also be intermingled within the Cossack regiments). These 'eastern' units often had a something of a reputation from enemies and allies alike, generally viewed as capable warriors but terrible soldiers, being undisciplined and needing close minding to prevent looting even when passing through friendly territory. Of little use in traditional cavalry roles, they instead served a valuable role as screening forces while on the move, and reconnaissance at the early stages of a battle.

I don't have anything too extensive for the Tatars or the Kalmyk, but for the Bashkir, I can say that they served extensively during the Napoleonic Wars. Generally numbering 500 men or so at the time of muster, 28 regiments were raised by the Russians for service during the conflict. Sources describe their blue uniforms as "in the native style with kaftans of blue or white cloth, trimmed in red, and white caps", and even though it might have been the gunpowder age, many continued to carry a bow and arrow to complement their firearms, which they could use with deadly efficiency when working close in. Although often a Russian officer would be in command, in some cases they would be led by a Bashkir, and being of Muslim faith, a mullah accompanied the regiment as well.

Not counting the numerous Cossacks, present at Leipzig, specifically, was the 2nd Bashkir Regiment and the 2nd Kalmyk Regiment, both as part of the 2nd Hussar Division, the Stavropol Kalmyks with the Cossacks of the VIII Infantry Corps, as well as the Eupatoric Tartar Horse Regiment and the Tartar Uhlan Regiment in the Combined Uhlan Division of mixed German and Russian units (I might be missing a few, the order of battle at Leipzig is immense. It wasn't called the Battle of the Nations for nothing!). Based on the style of dress of the rider in the painting, he seems roughly in line with this depiction of a Kalmyk rider from Hofschroer's book, but that is by no means definitive, as this image I found of Bashkir warriors of the period also seems fairly close, as does this, if not in color. Likewise some more plates which depict the variety of costume don't quite give a definitive answer. It doesn't help how in shadow the figure is in the painting, wearing what might be blue but appearing more purple in the lighting. I wouldn't want to take a firm stance on which, specifically, the painter desired to represent short of a statement by him, but certainly they were quite correct to depict an "Asian warrior" being present.

Baumann, Robert F. 1987. “Subject Nationalities in the Military Service of Imperial Russia: The Case of the Bashkirs.” Slavic Review 46 (3/4): 489–502.

Russia Against Napoleon: The True Story of the Campaigns of War and Peace by Dominic Lieven

Leipzig 1813: The Battle of the Nations by Peter Hofschroer

The Military and Society in Russia 1450-1917 ed. by Eric Lohr & Marshall Poe

Napoleonic Uniforms, Vol. 4 by John R. Elting