How does one create those often hilarious moving avatars or user logos?

Sorry, Fluther is not set up that way.

You just need an image editing program that lets you create and save GIF images and naimated GIFs. It can be a bit labor intensive, as you make them the same way movies are made, except you have to create each succisive image that moves just a little bit. You create a series of a few GIF images then save them together as an animation file. Depending on the image manipulation tool you use, you will need to read the details of how to name and save the separate panels into a single GIF animation.

Happy image editing. However, @AmWiser may be right. It looks like to me Fluther allows the uplaod of GIF, JPG or PNG avatar images but defaults to saving them to the JPG format, which does not support aniomation. So check that out before you go to any effort creating an animated avatar if you want to use it here.

they are just animated gifs with 2 or 3 images cycling through.
They are very irritating after a while