If you'll allow me to clarify, I don't believe Russian state media is behind the protests, but I do believe the Russian state media, and possibly other foreign media, are exploiting the situation. What Russian state media and the Kremlin try to do is identify very divisive issues and exploit them in order to further divide western societies.
We've observed over the past week that RT—Russia Today—has been reporting positively on the protests, and we've seen other proxy platforms doing the exact same, which legitimizes them and could perhaps help fuel them along. However, it would be incorrect to say that Russian state media is behind the protests. They just exploit them.