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The tilemap has nothing to do with the transition. Also that battlebackground was made by Harry for the community hack, it's not for public use without his consent :/
Oh, ohk, well then,
Lemme change my question,
How do I edit the image for the result in the gif?
Some info about layering transitions are given in the Tut called
'How to change battle backgrounds'
Absolutely not. Even the gif you posted, the bottom part is also moving the opposite direction as the top half. Seems like the BG was designed around this fact and the bottom half doesn't have a platform.
If you're looking to have the entire thing scroll in a certain direction, it will take more work than editing an image.