I am in the process of making a 'Game' using the Unity game engine which features pre-rendered 3d background images. Currently, we are using JPegs but I thought it would be nice to get some movement in them. I remembered Final Fantasy VII having really neat pre-rendered images with movement in them and researched it and they still look great. the images on this site are gifs but I'm not sure what compression would have been used back then, very possibly gif. I seem to recollect that animated gifs only update the area of the image which actually changes which helps with compression. My question is: Is there a modern equivalent to animated gifs that would allow me to create high-quality background images with small areas of movement in them?
The "modern equivalent" of animated GIFs is... MPEG. Or really, any decent movie compression format. These techniques are designed to spend fewer bits on static areas and more bits on moving ones. And better still, they're designed to be dynamic, allowing motion in any are at any time, rather than only in fixed areas.
So just make a short, looping movie of the "background image" you're trying to build. You may want to get into the details of the compression system you're working with, so that you can give lots of bits to the initial frame and fewer bits to moving areas.