That link may not work, sorry!! Try this one:
https://drive.google.com/file/d/1O8sr2YIBDT2Sn0yl-BQfj1ZVUDpacw18/view?usp=sharing
Thanks, that worked to reproduce the problem. Turned out to be a bit subtle: your puppet has three views, but only the Standing view has Draggable wrists. When stopped, you were seeing the Standing view, so dragging the wrists worked. When recording the Walk behavior was triggering the Left Profile view*, which has nothing draggable.
I figured this out by turning off the eyeball for all the behaviors except Dragger and then turned them on one-by-one to see when the problem started.
Easy fix: just tag the left & right wrist handles in the Left & Right Profile views with the Draggable tag.
*I think this is a bug because you have no Walk Position keyframes so it should stay on the Standing view; in any case it doesn't happen if you turn off the keyframe stopwatch, or have actual changing keyframe values