No Soviet republic had its own military units. This applies not only to to Asian republics but also to Ukraine, Belarus, to all republics. There was only one Soviet Army. That you see in it a lot of "Asian-looking" soldiers is not surprising: it was based on military draft and reflected the general composition of the population. People drafted in Asia could serve in Ukraine and vise versa.
This applies also to the so-called "Interior troops" whose stated purpose was to protect the boundary, but they were also used sometimes to suppress riots.
Only police (which was not related to the Army) was manned locally.
There were very few exceptions in the history of Soviet Union. For example, a Polish unit was formed during WW2 of the Poles taken prisoners in 1939 and other Poles.
EDIT. Few words on the comment of Felix Goldberg. KGB existed only since 1954 (as a result of re-organization after Stalin's death). Before that all interior troops
and border troops were controlled by various ministries.
(Basically this was the Ministry of Interior but it frequently changed the name). They all originate from VChK squads. Then it was GPU (1922) OGPU (1923) NKVD (Ministry of the interior) (1937), MVD (1946) MGB (1947), MOOP (Ministry of Protection of Public Order) (1962), and MVD again (1968). The main stated purposes of these troops were protection of order, boundary control, guarding important objects (rail roads first of all) convoy (prisoner camp maintenance) and construction work.
In 1954 border control function shifted to the separate agency, KGB. Interior troops were responsible for repression of popular uprisings and riots, deportation of peoples during WWII, and other such functions. Some of them also fought the Germans during WWII.
All interior troops were under the central government control, there was no
national units controlled by the republics.