The mass deportation Chechens, Ingush, Crimean Tartars and Kalmyks are examples I know of. I know one justification was for (supposed) Nazi collaboration, but as far as I know the Chechens, for example, didn't have nearly the level of collaboration as found in Estonia or Ukraine, and yet they suffered far more extensive deportation.
The second part comes from my understanding that the predominately Muslim central Asian Turkic peoples tended to see the politicized atheism of the Soviets as being the same sort of anti-Islam as found in the politicized Christianity of the Russian Empire.
The official reason was because of alleged Nazi collaboration, but as thousands of Chechens (and the others; I'll use Chechen here to mean all North Caucasus groups deported, but don't mean to imply anything aside from ease of typing) had served, and were still serving, in the Red Army, it was nonsensical. Granted there was some efforts by the Germans and local Chechen groups to work together, but that was not very successful.
I would also say that their religious views were not a factor. Keep in mind that nearly all of the Central Asian peoples (Kazakhs, Uzbeks, Tajiks, etc.) and many other groups in Russia itself (the Tatars being the largest group) were, and still are, Muslim. While many Chechens practised a hybrid Sufi Islam and traditional Caucasus beliefs, not really the same as the other Muslim peoples in the Russian Empire/USSR. Still, like I said, I don't think that played a factor, or at least a large one.
I would argue that the main reason stems back from the Russian Revolution and formation of the Soviet state, and even then one can trace it back to the first Russian attempts to pacify the North Caucasus in the early 1800s. The Chechens had always resisted Russian (and any power before them) control, and had actively spent decades fighting anyone who tried to establish themselves. This probably reached its peak under Iman Shamil, who managed to fight the Russians for decades before the Russians effectively won control in 1864.
But even then the Chechens were not willing members of the Empire, and so when the Revolutions of 1917 they broke free, nominally creating the Mountainous Peoples Republic of the North Caucasus (the name has been translated into a variety of ways, but that's the basic idea), which lasted for a few years. Once again the Russians (in the form of the Red Army) tried to come in and reassert control, which was costly and time-consuming. The Chechens proved one of the most difficult to pacify, even though the basic idea of Marxist-Leninism was quite compatible with their traditional way of life (in terms of property and the like).
So I feel that Stalin and Beria found an excuse to deal with a troublesome people that were not interested in being part of the USSR and had caused more trouble than they were worth. Keep in mind it was not the first time a Caucasus people had been ethnically cleansed: the Circassians were forcibly expelled (they refer to it as genocide; I won't get into that here) in 1864, while the Abkhaz were also largely removed in 1878. So there was a pretext on how the Russian authorities dealt with the unruly people of the Caucasus. That they could cite Nazi collaboration just helped produce a legitimate claim to the whole operation.
/u/blindingpain might be able to shed some light on this topic.