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1. Double click on the puppet that you want to add the mouth into (this opens it in the puppet panel)
2. from the project panel, drag the mouth puppet (or any other puppet) into the desired group in the puppet puppet. Note that once it's dragged in, you can reorder it within the group, but not move it into another group, so make sure it's in the one you want from the start.
If you have a part of puppet A (as opposed to the entire puppet) that you'd like to use in puppet B, you can right-click on that part (group) in the puppet panel and choose Make Shareable. It will then show up in the project panel, and you can use it for step 2 above.
Sorry, not sure where the example puppet you're talking about is, but DaveM will likely chime in when he's back from vacation.
Most examples are here: Character Animator Examples
or here: Okay Samurai Puppets
thank you. Huge fan of your work and dedication by the way!