The ASS format has some basic animation with the \move command (moving subtitles using linear interpolation) and the \t command (changing various properties of a line with the possibility of quadric interpolation). This takes care of some cases, but is often insufficient for complex stuff.

Aegisub however has good automation support, and has (had?) a pretty active scripting community. Scripts can do a lot of stuff and often has simple dialog-box UIs. The base scripting support is in Lua, but a lot of scripts are written in a Lua variant called Moonscript. I've never seen it used anywhere else. And, as was mentioned in a sibling comment, there are scripts for importing motion-tracking files into Aegisub.

The ASS format lacks support for a bunch of stuff that people want to do, or the native support is deficient in some manner, so scripting is very important for Aegisub. For example, a line can be given a color gradient by make a one copy of the line for each shade in the gradient (this can be 100+ times). Each copy is then given a different color according to the gradient, and a \clip command is used on each line to only show a 1-2 pixel wide slice of each line. If all those slices are lined up properly and the lines are displayed simultaneously, it gives the impression of being a single line with a color gradient. Tricks like this can actually cause performance problems in rendering the subtitles!

I think this is an interesting little programming niche, but there isn't much complex typesetting being done these days. Tastes have changed, and due to licensed releases there's less need for fansubbing on all but the most niche titles.

The subtitles are basically adjusted every frame but you don't necessarily have to do it by hand. You can for example generate motion data with Blender or Mocha and import that into Aegisub (the software that is used to create the subtitle files).

https://unanimated.github.io/ts/ts-mocha.htm

https://github.com/TypesettingTools/Aegisub-Motion/wiki/Appl...

Before newer tools automated the process by tracking motion of a target, folks wouldn't have to adjust them frame by frame per se but would instead annotate the subtitle script with keyframe times and positions. The video player would know to interpolate between those annotations.