How do you plan on handling this? I think that this may come up in many magical games, as well as scifi/space opera games where there are holographic projections.

Currently I'm working on a post-Voyager Star Trek campaign, so holographic crew with mobile emitters are definitely possible. But also for the illusion magic when things try to attack a mage's mirror image type spell, or attack an illusion of a dragon or something.

Should they just be immune to damage, or some other narrative effect? Or take strain to represent dwindling energy of the illusion/hologram?

What are the community's thoughts?

Strain seems the simplest option.

This is an issue in every rpg. To an extent, it depends on the narrative source of the illusion and the setting. In some settings, an illusion is every bit as dangerous as the real thing and will adapt itself to simulate damage it takes until you realize it’s not real, in other settings an illusion is completely intangible and incapable of dealing or receiving direct damage.

Having the construct take strain is certainly one possibility. I could also see a GM using advantage generated by the players to have their characters realize an illusion isn’t real.

In a Star Trek setting, a hologram with a mobile emitter should probably just be defined as a type of species with some special effects to represent its hard-light nature, but other than that it’s just a person. They are as solid as anyone else, though some creativity on the critical hit table might be in order.

Speaking of, once they reverse engineered the mobile emitter, do you suppose anyone rummaged around for Moriarty and actually freed him like Picard promised? Could be an adventure there, his reaction to being duped and what he does next.