I have plenty of experience of GIFs but none of FITS files so I am on very unsure ground here. A Google search, however, tells me that the downloadable program realConverter will convert FITs to JPGs or GIFs. Whether this runs with Linux I am unsure, probably not.
Once in this format they may be processed as animated GIFs. I use Photoshop for this but there are plenty of free GIF makers on the web.