Properties such as hair and skin color can be changed without affecting the armature-based animations at all – these are materials properties, which can be animated, but separately from the armature unless you decide to purposely link them together in some fashion, say with drivers.
Clothing is another matter, and depends on how you’ve implemented it. Garments that are armature-animated, or actually part of the character model mesh, will probably need to be handled as different animations for each change, but that depends on how they are designed. The figure animation can stay the same if the clothes are separate meshes, though. In that case it’s a matter of adapting the clothing animation to the existing character model animation
For cloth-sim clothing, each garment mesh will need its own bake for every animated sequence. These are almost always separate from the character figure mesh, so the animation for the full figure need not change with clothing change, just the cloth bake. The physics cache will contain baked frames for each cloth item done this way.
If a combination of methods is used for clothing, then all these comments will need to be adapted based in the specifics of your character and its clothing.