Dealing with different skin color textures (Model Texture variant)

I'm wondering how to best deal with texture variants on the same model. Let's say we have a character in a business suit. Many games allow for character customization, like choosing from different shades of skin color, or the color of the suit. How could this be accomplished in the most efficient way?

Using one texture for each variation would mean that I'd have to have a blue suit with light skin, a blue suit with medium skin, a blue suit with dark sin, a black suit with light skin,... Not very efficient, there has to be a better way.

It would be possible to make the character a collection of body parts. One for each hand, one for the head, one for the rest of the body (suit and shoes). These could then get assigned their own textures.

However, it would also create problems with the character's animation because for example, a hand could only be attached to the main body's hand bone, but any finger bone movement couldn't be done without the fingers also having to be seperate models. Also, in case of the head, the head model wouldn't be smoothly connected to the neck/torso