Have you tried searching "bone animation"? Usually a 3D game with an avatar system has an animation library for the humanoid skeleton that underlies all characters. Things like hair, capes, tails, or wings may have their own animations, but they have an anchor point or 2 or 3 where they attach to the character skeleton.

Character Creation and Integration for Video Games

...What I am unsure of is the process from a finished model to a programmed 3D character...

Have you tried searching "bone animation"? Usually a 3D game with an avatar system has an animation library for the humanoid skeleton that underlies all characters. Things like hair, capes, tails, or wings may have their own animations, but they have an anchor point or 2 or 3 where they attach to the character skeleton.

This is pretty spot on about how character models work.

Here's a general overview.

The way character models work is you first start off with the character mesh, then you rig it with a program. The idea is that you put a skeleton underneath for manipulating/deforming the mesh. Most modeling programs will have some sort of rigging capability. Then that rigged model is exported and put into the game engine.

Game engines usually have some sort of library/functionality for working with bones/joints. This varies from engine to engine, and to answer this, it'll depend on what engine you're using.

Hope this helps!