While it appears you have fixed movement animations on generic avatars, movement animations on humanoids are still non-functional.
If a custom movement animation has keyframes that aren't humanoid flexes (and thus aren't synced by the network IK), the animation will appear to play correctly locally but remote players will not see the animation.
Example: avtr_86cbeb8a-1886-4449-8e81-478dccdaa43f
This avatar uses blend shapes for the movement animations. Locally running appears to work correctly. Other people will just see you slide around playing the idle animation.