This is a pretty wide stretch but think about how a door handle works on like a commercial building where you push down on it on one side and it also moves on the other side. If you had two of those handles on the door pointing in at each other and covered them with fur they would look like eyebrows that move up and down when moved the handles on the other side. Maybe this gives a better idea of what I am talking about.
YouTube VideoSometimes instead of the action you see above the eyebrows are mounted on a post and it moves up and down in a slot in the head.
Animal has it right. If you want to operate the mouth with your one arm then you create a dummy one on the puppet. It is normally strung to the chine or upper chest so that it has some movement and does not just hang to the side.
Sorry not a lot of diagrams of this out there.