Z-sorting with Cycles?

are your : neutral eye brow angry eye brow like separate PNG pictures in your actual solution?
if so, yes… technically you can, read my post. the mix node if it’s 0 it takes the first picture if 1 the second picture plugged into it, so you are switching between texture, and you can animate that switch value.
but i was saying not sure if it’s the optimal solution for you cause i don’t now much.
if the whole face is one picture simpler, if the eye brows are separate to the mouth and stuff still can but you will end up with a complicated node tree
i believe NLA is enough for that type the animation but if you want the action editor, but be aware that the animation is on the material not the object:

or you can keep your actual solution and cheat by moving the planes a little bit to avoid Z-sorting issues and disable their contribution to Shadow so they don’t cast shadow on the model,