Cartoony Eye FFD Space Warp

Make an eye with some rotation animation:

Make an FFD Space Warp ... position it around the eye ... set the number of points as desired ... I used 3x3x3:

Bind the eye to the FFD Space Warp  so your stack looks like this:

 

Select the FFD box itself [not the eye] and animate the control points of the FFD Box ... ... you will retain the rotation of the eye as it deforms: