Most of the time it's much simpler than we imagine.
I think it's static geometry with an animated texture (procedural or pre-rendered as a looping sequence of frames) that is moved with the texture matrix so that it slides along the geometry.
The vertices at the end of the geometry strips have zero alpha, so the effect disappears into the air.
So that's essentially an animated spritesheet on a quad? That seems plausible, but also potentially expensive to render. Some of those animations are pretty detailed and long.
In any case, I am going to give this a shot.