Here are two animations illustrating the deformation process. Limit sets are drawn by lim, written by Prof. McMullen. The two pictures are showcasing the same process, but supporting hyperplanes for the convex core boundaries are added in red in the second picture. A paper explaining these pictures can be found here. For a much shorter excerpt focusing solely on the example see here.
Warning: the pictures are very large in file size, ~70 mb for each animation. There will be significant load time before the animation starts.
Download a .gif version of the animations without or with supporting hyperplanes. They are about 70 mb each.
Below is an animation showing just the quasi-Fuchsian groups in the process above. The supporting hyperplanes on the side corresponding to the acylindrical manifolds are drawn in black, and for each frame, a pair of supporting hyperplanes on the other side of the quasi-Fuchsian group are drawn in red. Other supporting hyperplanes on this side are in the obrits of these two under the quasi-Fuchsian groups. This animation is a visual representation of the skinning map. Note that at the end the two red circles intersect in a positive angle. This reflects the fact that the image of the skinning map is bounded.
Download a .gif version of this animation. It is about 13 mb.