Thanks for the puppet. I found two problems.

Your "Head" layer had handles added for Left Eyebrow and Right Eyebrow (and Nose). I deleted these as the behaviors were attaching to these handles, not the child layers.
Second, everything is independent! You need *some* artwork in the Head layer for the children to attach to. So I turned independence off the face background. (Might want to turn off the hair independence too maybe.) Notice now the Head layer has a yellow mesh. Before the child meshes had nothing to attach to which can cause problems.

After that, I decreased the Head Position Strength from 100 to 5, so it does not drift away as much (you can try zero too).
I think you will need to do more with the hair. E.g. the rear hair perhaps should not be independent, and the front hair has independent layers inside an independent layer - again, the parent hair layer will have not artwork in it (the mesh will be zero size) which is probbably not what you want. I suggest removing independence from the rear hair and parent front hair layer. Basically avoid independence inside independnce unless at least one child is not independent.