The secret of the controls for such a puppet would be found in a bicycle shop.

Look at the brake cables. It is a flexible tube that has a wire running through it that can be pushed and pulled to move the puppet. It could be that the Woodbabies use an adaptation of this but the concept would be the same.
Your first step would be to design the body that is under your fabric that could be manipulated by the brake cables. I wish I could give you specific instructions but there aren't any out there really.
