It looks like I misunderstood your first video. Apparently, they are using “dynamic environments” which is just an animated HDRI world. (They may be using a sphere mapping/matcap technique, which only maps a hemisphere of the world, rather than latlong maps.)
You can load an animated texture into your world nodes just fine and it works. The only problem is actually getting your hands on an animated equirectangular or sphere mapped HDRI. You could potentially render your own.
You can convert equirectangular to sphere maps if you want, although it’s kind of arduous, and you can convert a sphere map to about half of an equirectangular map if you want.
Edit: An easier way to get what you wanted would be just to animate the mapping of the world. Somebody watching would have a hard time distinguishing between a truly animated world image and a world image that with a few rotation keyframes.