Keep in mind that the history of these ethnic republics with Russia has been been volatile to begin with, especially in the Caucasus. For example, Chechens are usually organized by clans called teips. But not only that, Chechen is a Northeast Caucasian language, having many unique pronouncians that are not found in most other languages. Chechens value freedom and nature. One of the more famous people is Zelimkhan, who was an outlaw who stole from trains and banks from Russians and gave to the poor Chechens, kind of like a Robin Hood figure. Of course, I would imagine many Chechens hate Kadyrov.
Now, the Baskirs, who are a Turkic group (different group from the Turks in Turkey), have not really fought Russia. However, I think because of the war in Ukraine, they see a weakness. But I am betting that a big reason why there are such big protests is that a lot of the people thrown into the war in Ukraine comes from the ethnic republics. There are more than 200 ethnic groups in Russia, particularly on the Asian side. They don't want ethnic Russians to die there. Also, this contradicts Russian World.