Photoshop :: How To Map / Morph One Image Completely Into New Shape

I want to take a map of a country and change the shape of it to fit exactly into another complex shape (e.g. a puddle of water, an animal outline, etc).

To be clear, in doing this I don't simply want to mask the original map image to "cut out" the chosen shape, I need to keep the entire map and distort its shape into the new shape. I'm also not talking about using displacement maps to "wrap" the map around a second object. I guess it's a "shape morph" I'm after, which leaves all the original content and just changes the boundaries.

I know I could use the warp tool, but this requires an incredible amount of time to manually change images to exactly fit a new complex shape. It also means every time I want a new shape it has to be manually done. Given that the warp grid only has 9 squares, this makes it even more difficult to get a good match to the new shape.

For example to choose a few points on the original to map to chosen points on the second shape, such that the software will then interpolate the points in between.