So if a hero dies, do their cards under La Capitan go into their trash, or do they stay there?
All their cards are removed from the game when they're incapacitated, and the crds under La Capitan are still their cards, so I'd say the cards get removed.
So the Captain killing her enemies removes her own protection?
The rules agree but it seems counter to the idea of putting cards beneath her to me... unless it follows the same idea as cards under Savage Mana, that it is a hero card and has no text, but (wild supposition ahead) as it has no text it is not associated with any specific hero deck. But that is really stretching it.
EDIT to make clear that it is merely my theory and the not-associated-with-a-deck portion is not an official ruling (the bit about it being text blank is official)
Thats! The ruling i was looking for. I knew it was some where, but this happened to us against La Capitan the other day and it was my opinion that they should stay under her. Smafdis was that all hero cards are destroyed when a hero goes incapacitated, so too should his cards under La Capitan - but i /thought/ there was a ruling somewhere that said its not associated with any specific deck
thats how I see it.
I look at it this way: The cards being used to protect her are her way of using the heroes' "possessions" against them as pressure/leverage. The heroes take their things back in the struggle instead of actually doing damage to her. When a hero is no longer in the fight, the heroes are no longer worried about that hero's possessions, so they are no longer protection for La Capitan as the heroes aren't trying to get them back.
Yes, some of these "possessions" are one-shots, etc., but there are other thematic ways to explain/justify. Just imagine it's a comic and La Capitan is using the heroes' resources, concepts, and ideals against them until they can "get them back".
That's just my opinion, though - feel free to come up with your own! 
"I could 'rescue' Bunker's Omnicannon but he's not in the fight anymore so I think I'll leave it here. With the villain. For later use against us."
doesn't not sound particularly heroic (or friendly!)
Yes, some of these "possessions" are one-shots, etc.,
Wait, why would there be one-shots under La Capitan? Her ability is "whenever a hero card is destroyed by a villain card, put it under this card". One shots aren't destroyed when they're played, they're just played then put in the trash. And they're certainly not affected by a villain card.
She's got that "take the first card from the top of hero decks and put it under" thing
Yeah, what celette482 said. In my experience, that's been how she usually gets the cards from us, anyway.
This is something I asked Christopher in November when he came to Indianapolis for Infernal Relics. He had set up various tables to play test the villains who had been released to the testers at that point. I had gotten La Capitán. He told me that when a hero becomes incapacitated. all of his cards are removed from the game, even if they are underneath another card and have no text. That would include La Capitán and Savage Mana. (I would include Omnicannon, but only Bunker cards can go under there, and if Bunker goes, so goes his Omnicannon.)
That sounds official enough to go in the Clarifications doc. Thanks.
You're definitely right buddy. Removed, but not totally lost.
[EDITED TO REMOVE SPAM LINK -- arenson9]
Except you forgot that the '[' is still linked. Unless you are trying to preserve this for some strange reason.
[FIXED – thanks – arenson9]