The lamp from Circuit Condition Lamp seems to add a custom lamp mask, which doesn't move with the lamp entity.

This needs to be fixed in the other mod. See the note in https://github.com/hgschmie/factorio-even-pickier-dollies/blob/main/README.md

Thank you for pointing it out! I was scratching my head over how some modded entities leave their stuff behind, most notably the invisible inserters from Ammo Loader+ mod.

same there. EPD is not part of the game mechanics but uses some clever tricks to move things around. If a mod uses "hidden" entities for functionality, these mods need to support EPD through the remote API.