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The main concept is that I think of the facial features as 'stickers' that I place on the main body texture's face. I use multiple UV channels and textures to simplify the process. I think of these textures as 'sticker sheets', and use bone drivers to select the 'sticker' I want.

Are the elements you're moving in this anim part of some 3d UI of the software or are they polygons you created as a way to "remotely" control the bones (or however this is achieved, I don't know this in-depth)?

The trick for the eyes is the bones inside the face that drive the UV offset for the pupils. Using bones for this has the benefit of providing all the IK functions, which also works in the engine later on. Calculating the offset is just trigonometry. (or you just eyeball it🥁)