We should use videos instead of animated GIFs, animated PNGs or animated WebPs for the following reasons:

• Better compression, which allows using higher resolutions and framerates.
• Support for playback controls.
• Support for audio if needed.

I have a WIP branch for this: https://github.com/Calinou/godot-docs/tree/add-video-support

However, this is blocked on sphinx-contrib/video#33, as we'd have to include all videos in a top-level _static folder otherwise.

Edit: The above issue was resolved by sphinx-contrib-video 0.2.1.rc0.