That’s changed and men of an age eligible for the military are explicitly not allowed to leave. If they manage to bribe their way out, their families will receive a 5% tax - on top of any other taxes.
In Russia, as far as I know (and I have close and active sources of information), there is no change in freedom to travel unless you are under a formal mobilization order or are already serving in the military. Many Russian men regularly travel abroad for any reason at all. There is no special or extra-constitutional regime in place.