Your UV set is after the skin cluster in the inputs? If it is, you'll need to move it to the bottom of the inputs. I know this tread is very old now, but it comes up on Google if you are searching for help with this issue. It does seem to be the UV input being above the Skin Cluster input. I could not move Skin Cluster, so instead I deleted its node and re-binded the skin. That way it is now above my UV inputs and the UVs stay put when animating! Hope this helps anyone else with this issue!