Hello!
The boy's mouth is made with the same pieces the girl in the gif below would have:
upper lip, lower lip, teeth up, teeth down, tongue, mouth isides.
The mouth is controlled through two paths, one for the upper lip and one for the lower lip, and it overall should have around 4 bones to control each side of the mouth with path constraints.
I hope this clearns things out! if not, you may be in luck, as I'll actually rig a similar mouth very soon, and I'll likely record the process!