When I tilt the head toward the shoulder the eyes, nose, and mouth move out of the bounds of the face. This is different from when the pupils move outside of their bounds.
I have two characters, one works the other doesn't, so my guess is that I missed a minor step somewhere.
This is usually from making stuff independent - adding the + in PS/AI or the crown icon in CH - when you don't need to. In the puppet panel if you see crowns next to stuff like left/right eye, nose, mouth, face background - try removing them. The only plusses/crowns you really need are for the pupils, eyebrows, and top character group.
If that doesn't work, feel free to post or DM me an exported .puppet file and I can take a closer look.