As I was trawling through different pictures and videos regarding the Eastern Front in WW2 I couldn't help but notice a lot of Asiatic looking men in The Red Army and I became pretty curious about the involvement of Russia's pretty diverse ethnic groups in WW2.
Anthony Beevor's "Stalingrad" talks of a huge amount of asiatic Russian reinforcements being moved into the Stalingrad front to shore up the Soviet position. The account states these were recruited in the eastern Sovet Union and sent westwards as soon as possible to the front by way of the trans-siberian railways, pausing for a period of several weeks to carry out basic infantry training in the vast steppes of central Russia, wherupon they would be moved to marshalling points close to the Volga crossing to be equipped and moved across the river to replace exhausted and depleted Russian units holding out on the western bank. This slightly rushed nature of deployment led to what one might call a variable quality of soldier in terms of equipment and training, not a problem unique to Asian Soviet forces, but exacerbated by the enormous distance travelled from their homelands and the feeling of alienation that resulted, especially due to the fact that not all these Asian soldiers could speak Russian fluently, or at all. The less refined nature of these units sometimes resulted in accidents, occaisonally fatal, when they had to operate weapons or munitions unfamiliar to them. Several accounts state casualties were often caused by hand grenades being used improperly; one such account mentioned an Asian soldier attempting to dismantle a hand grenade he believed himself to be familiar with, whereupon it exploded in his lap and killed him.
Source - Артемьев А.П. Братский боевой союз народов СССР в Великой Отечественной войне. М.: Мысль, 1975. С.58.
Quick translation: ethnic % of Rifle Divisions (Soviet infantry). Columns of the table - by January 1st 1943, by April 1st 1943, by July 1st 1943, by January 1st 1944. Last column - percentage of respected ethnicities in USSR population according to 1939 census.
The rows you're after - Uzbeks - 7, Tajiks - 8, Kazakhs - 9, Turkmens - 10, Kyrgyz people - 11, Tatars - 18, Bashkirs - 21, Kalmyks - 22, Buryats - 26.
Quick calculation gives ~10% of Soviet's Asian ethnic groups served in Rifle Divisions. Which more or less reflects ethnic proportions of Soviet population at that time.