My 2 cents :
For Displaying and animating :
- https://github.com/TomDannert/GIFImage
- Skia4Delphi !
I do not advise you to use http://www.raysoftware.cn/559.html : It unfortunaly not work on all gif, and on recent compiler, it does not pass range checking.
Tom Dannert's version (https://github.com/TomDannert/GIFImage ) is a far better start if you want to write a Gif Writer.
It is right that, under FMX, full cross platform (linux and mobile included), I do not have direct solution to write GIF.