I don’t really frequent the forums, because Blender has such a large archive of already solved issues that I usually get along fine.
However, I now have an issue that has completely stumped me and I cannot figure out what it is. I made a very (!) low poly character to test out some things and I have some very weird UV jumping issues in the internal render. The issues do not show up in OpenGL.
It looks like bad skinning, likely related to the lowpoly nature of the character. You might want to create a bug report. Strangely though, the second animation looks fine.
Yes, the second animation is from the viewport, where no issues occur. Perhaps I should try what the UV does in Cycles. If noone else has any ideas I will file a bug report.