Just use layers?
The whole idea of lenticular graphics is that each "distance" (background, middle distance, foreground objects, etc) is put on a different layer, then several versions of the image are generated with different offsets. For example, the range of images might start with the background moved two units to the left, to two units to the right, with half a dozen steps in between. Other layers are offset at different amounts. Sometimes, different images are used at each step, to get more realistic perspective, or to add animation.
After that, specialised software combines them into one print, and a specially constructed plastic sheet is lined up on top. The grooves in this sheet ensure only one image is seen from each angle. However, each eye will pick up a slightly different image, giving the illusion of depth.
To create the animation at the end, you can see that as foreground objects move to the right, background objects move to the left, and vice versa. With careful positioning of each layer, perhaps using Layer Comps to save each state, you should be able to create all the necessary frames.
