So, the most important factor when it comes to institutional race in the USSR is when you're talking, because like most things in the Soviet Union, leadership mattered

At the very beginning of the USSR, you can basically draw equivalents to affirmative action in the US. The early Soviets drew up a policy called коренизация/korenizatsiya (rough translation; nativization) which had two major objectives; number one, curb what Lenin called Great Russian Chauvinism, and number two to promote national minorities and therefore make them good communists. To that end, every organ of the Soviet state had a national version and a Soviet Republic version (except for the Russian SSR, which was considered to be a federation within the Soviet Federation, hence where Russian Federation comes from now). Schools in the ethnic SSRs and ASSRs taught largely in the national (as in narod) language, and Russian officials in these Republics, while not purged, were encouraged to learn the language

Here it's important to note that officially race was not used that often in Soviet documents. "White" or "Asian" didn't really come into play. Instead, in Imperial and Soviet Russia the term used was Nation; Narod. It was relatively strict; one couldn't really "switch" nation after all. Your nationality (not Soviet, but Russian, Uzbek, Tatar, etc) was its own line on your ID, passed down from your parents. If you came from a mixed nation family, you had to chose one or the other for most of the USSR

So then we get to the Stalinist era, and things go very poorly for minority nationalities. Many of the cadres of educated officers and administrators created by Korenizatsiya (especially the Ukrainian, Kazakh, and Caucasian Cadres) were purged, on charges of "weakening the bonds between Nations, wishing to disrupt the USSR, and bring back Capitalism". Other posts have gone into what happened to Jews, Chechens, Tatars, etc specifically so I won't, but basically bad shit happens to a lot of national minorities. In this period there certainly were ethnic minorities in leadership roles (Stalin himself being the most prominent as a Georgian) but there was a distinct shift in Soviet language from being a union of brotherly nations to the Russian nation being the Eldest brother, and therefore the leader

Khruschyov takes over and the thaw begins. A lot of the most repressive pro-Russian policies (such as the wholescale purge of national cadres, increased use of the Russian language, direct rule of minority ASSRs by Russians, etc) are ended. It's about this time that large ASSRs start to be ruled by members of their nation (Tatarstan most predominantly, as they end up being the Russian Federation's biggest regional pain in the ass after Chechnya)

After Khruschyov, his reforms are somewhat reversed in turn. Brezhnev and the CPSU tries to deemphesize nationality as a whole. It's around this time in the early 70s where they come out with the idea of the Soviet nation. Essentially, the idea is no repression for racial differences, but everyone would speak Russian. To that end, the Russian language goes from being merely a subject at every school to being the main body of education for a large chunk of the USSR. After that is less my forte, but Gorby in turn re-reverses those reforms to something closer to what Khruschyov had, but that's getting close to the 20 year rule

I'm away from home right now so I don't have my books with me, so I'll add some sources later

Although speaking only about the Army, this might be of interest.

Related question: I'd be curious to hear about anti-Semitism in the Soviet Union. It was in principle outlawed shortly after the Revolution, but Soviet campaigns against "rootless cosmopolitans" and systemic discrimination against Soviet Jewish scientists are difficult to square with this position.

/u/kanadskiy has a good summary written up there, so I'm not going to try and do the same. I will, however, provide some sources:

There is a growing number of works that look at Soviet nationality policy, though a considerable amount mainly focus on the early era (pre-1936, when the so-called "Stalin Constitution" was implemented"). Of these the paramount has to be Terry Marin's The Affirmative Action Empire: Nations and Nationalism in the Soviet Union, 1923-1939 (2001). This is by far the standard work on the subject, and looks at how the Soviet Union first implemented the korenizatsiia system, and then ended it. A related book is Francine Hirsch's Empire of Nations: Ethnographic Knowledge and the Making of the Soviet Union (2005), which looked more at how the Bolsheviks created nationalities and national identity. A broader look would be Ronald Grigor Suny's The Revenge of the Past: Nationalism, Revolution, and the Collapse of the Soviet Union (1993), which was one of the first books to really offer an idea that national identity was fostered, not repressed, in the Soviet Union. Of course this goes with Yuri Slezkine's monumental article, "The USSR as a Communal Apartment, or How a Socialist State Promoted Ethnic Particularism." (Slavic Review Vol. 53, No. 2 (Summer, 1994): 414 – 452), which is one of the authoritative articles as well.

There are also several books that look at various ethnic groups within the USSR, and to list them all here would be a daunting process (there were roughly 120 various ethnic groups in the USSR, and while not each one has been studied, as far as I'm aware, enough have been to make this a challenge). But everything from the Uzbeks, to the Turkmen, to the Georgians, Jews, various Siberian peoples, even the Russians, have books that look at their interaction with the USSR as a whole.

You will notice that I did not include more modern events. That is both because the field is considerably less studied than the earlier era, and my own knowledge is lacking compared to the 1920s and 30s. However I could be prodded to list some titles in that area as well, I just am far more comfortable with this aspect (I wrote my MA thesis on one of the nationalities and their experience).

/u/Georgy_K_Zhukov already linked an excellent answer but you might also like:

How was the Black Panther Party viewed in the Soviet Union, the People's Republic of China and other communist countries? by /u/kieslowskifan