Teeth won't move properly with face rig

I created a character, and rigged him, and for the most part everything works fine. There are some tweaks still needed, but my biggest problem is the teeth. Like the eyes, they are a separate mesh from the head, and are parented to the head bone. They move with the head, but unlike the eyes, they move slightly relative to the head, so when he moves his head, the teeth are out of place.
I’m pretty sure they are being influenced by other bones, but when I go into weight paint the teeth are red, and they’re blue when I choose any other nearby bones.
I’m certain others have had the same problem, but I haven’t been able to figure out how to fix it.
Any ideas?
Before moving head the teeth are in place

After moving head the teeth have shifted

Okay, I arrived at a solution, which I’ll post here in case anyone has a similar problem. I added (with Ctrl-J) the teeth mesh to the body mesh, and to the head vertex group. Also, I noticed the chest bone was influencing the lower portion of the face, which was screwing up the rig. You can see the improvement. (I just need to add the appropriate materials to the teeth.)