Destroyed reward equipment

What becomes of reward equipment (Chekhov's Hairdryer, etc.) if a card (or combo) destroys the equipment. Does it still go under the incap heroes in that play area, for later reclamation?

If it's already been claimed as a reward in a character's play area, it would go in that character's discard.  It would only go under a character's incap if it's destroyed due to that character being incapped or if that character is incapped while it's in their discard/hand/deck.

Oh, you fish rewards out of the deck/trash after an incap? That'll help with getting some of them back, in that case...

Yes.  When a figure is destroyed, all Objectives go to the bottom of the Objective deck, all of that Character’s cards go back in the box (unless you’re playing another version of that Character, of course), and any Rewards your hand/play area/deck/trash go under the Incap for that Character.

Once you've earned them, Rewards basically count as your cards for the rest of the game (I think you pretend they have your card back rather than the Mission details, since they are no longer Mission cards but I don't think they count as Character cards), so if they are destroyed then yes, they go to your trash. And as mentioned above, if you are incapped then all your Rewards go under the incap card and you can choose to retrieve one as your pre-Start-phase action on each of your future turns.