Look at the former Soviet republics that helped Russia win WWII by supplying it with an almost limitless supply of soldiers. All of them have refused to allow their citizens to fight for Russia in Ukraine.

And if you know your history you'd know that Russia had to pull out of WWI because the country literally ran out of men during the conflict.

Their political influence, even with right-wing politicians who might be skewed their way, will be limited. And could change at any moment as we've seen with Orban.