In a 1v1 scenario, I'd give both sides the Big Boss Comeback rule, regardless of whether they're actual Bosses or players, in which case they are removed from play temporarily, but not Taken Out.
In a 2v2 scenario and above, the target is immediately dunked into the Edge and Taken Out, but their allies may Rescue them to get them out on the following player turn, benefit from Heroic Spirit so their turn isn't wasted, etc.
I haven't played much with Edge-based styles, so I can't tell you more.