Thanks! This reference helped me figure out the source of the problem.
Using the virtual shadow map cache visualization, I could see that the tip of my weapon was casting a shadow into a VSM page that was outside the constantly updated page around my character (visualised in blue). The lingering shadow fragments wouldn’t get invalidated away until I walked near them again.
The Fix: Check Include Component Location Into Bounds for the skeletal mesh casting the problematic shadow.
You could also try simply increasing the bounds scale of the thing casting the shadows (e.g. from 1 to 1.5), but IMO this seems like it’s stepping around the problem rather than addressing it directly.
Probably worth experimenting with those other bounds-related checkboxes too if your problem is a bit different to mine.
There may be different performance implications of one or the other method, I’m not sure.
Similar answer with same conclusion(s) here: Can someone explain why the shadows are trailing? - #6 by cgnrat