For the motion part, there is already an extension to gaussian splatting for moving stuff: https://dynamic3dgaussians.github.io/
You can also just string a bunch of them together to create an animation:
https://twitter.com/8Infinite8/status/1699460316529090568
For relighting, there are lots of NeRF variants that do this -- it should be possible to optimize material parameters for the splats.