This should be really easy to do using blender and most free photo editing apps (gimp or krita for example).
First, separate the foreground image from the background in gimp and export out a png with a transparent background (gimp has a tool dedicated to this):
Next import that image into blender as an image plane (shift + a > image > images as planes). You’ll probably want to select emit or shadeless in the bottom left of the import window so that the image doesn’t need to be lit by the lamps in the scene.
If you are using eevee, you’ll need to set the blend mode for the material of the plane to alpha blend:
Now, subdivide the image a bunch of times in edit mode (don’t use the subdivision surface modifier).
Switch back to object mode and click the green object data button in the property editor. You should see a section in the property editor for shapekeys. Press the plus icon a couple of times to add a basis and key 1 shapekey:
Set the value for key 1 to 1, then switch to sculpt mode. If you use the nudge brush on the plane while key 1 is selected it should just make those changes to key 1. You’ll be able to drag the value slider for key one to have the image morph between the way it normally looks and your sculpted changes, and you can right-click on the value to insert keyframes and animate these changes.
Here is an example I made with a cc0 licensed image of a dog I found. I exported out a frame sequence, then opened that sequence as layers of one image in gimp, then exported it out as a single animated gif:
Here’s the gimp document and blend file I used:
dog.xcf (928.8 KB)
dog.blend (2.0 MB)
This is a simple example, but since that is an object in a 3d scene I could move it around just like everything else you create inside blender. If the animation you want doesn’t need to be any more complex than this example, then you should try using the animation feature that is built into gimp’s warp transform tool (aka “liquefy”):
Edit: Sorry if I explained a lot of stuff you already knew. I sort of missed some of the stuff you said in your reply. You probably know most of this stuff already since you said you make your own gifs and know how to model.