Animation Coloring

For my next art project, I've decided to color the wingless version of this free animated icon, but I've never done anything involving animation, and I can't figure it out. My main problem right now is that whenever one of the layers is made invisible, the horse has blocks of blank white on it. I tried looking up a tutorial on coloring animation, but couldn't find anything that seemed to address it, so I could use some tips. :3