The most simplest way of doing this is to make Sub State Machine. So based on the the skin , it can enter in to the appropriate sub state machine.

Each sub state machine will contain all the required states and transitions which your character currently has. So in short you need one more layer on top of the current animator controller, which decides which sub state machine is to be played based upon its skin.

Basically you just change the sprites through code depending upon the skin type. You should not put the code in Late Update as shown in the tutorial as its just for demonstration purpose. Call the function only once when you want to update the skin.

You can find the complete tutorial in this link - https://youtu.be/HM17mAmLd7k?t=1609

And the entire source code for the tutorial in this link - http://goo.gl/6oSzDx