How is this moving mouth effect done?
If I had to make it, I'd do something like this: 


The face has to go on one piece of paper/cardboard, and the lower jaw is a separate piece. The strip of paper that is attached to the lower jaw, will go through the hole just below the face, so you can hold the strip and move it up and down without your hand being visible.
A large chin can cover the hole, when the mouth is closed. When the mouth is opened, you'll see a part of the hole.