Sorry for not responding sooner - it's been a while since I've gone on the forums.
It wasn't that technical, but it took a long time to do. The hardest part was making the edges of the continents, oceans, and clouds blend together, but I made this a little bit easier by translating the map so that the seam was in the middle and later making some clouds cover up anything that still looked awkward. Because redrawing the edges of the continents takes so long to do well, I did this on a small map, but there is probably some way to make it be practical for a large map.
I saved the completed map, used Shape3D, made the background and atmosphere, and then flattened and saved the image. Next, I unflattened the picture, replaced the globe with the map, used Shape3D with an increased y-axis rotation (by default, axis 2), and flattened and saved that image. After repeating this a lot, I had images for each part of the 360° rotation, which I animated using the Animated Image plug-in.