Well, ultimately, The General would remove the wings from any person he killed. Once that happens, any kind of special immortality or healing factor a character has is no longer relevant.

As for TPs and Elites, they may do the same thing. Or just leave the character for dead, in which case the character's healing factor/immortality can kick in. Generally if it's a death that that character can bring themself back out of in a few hours, they won't have to worry about a DP. But being exploded, vaporized, or something like that would obviously be another story.