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Well, it's related to Time-Delayed Death and the Doom Standard Status Effect. I'm not finding a specific Video Game Characters or Boss Battle trope, though.
We do have Time-Limit Boss, but that only appears to apply to bosses. Nothing about mooks who need to be defeated quickly.
TV Tropes's No. 1 bread themed lesbian. she/her, fae/faerHm. I'd be inclined to say that something like the Tonberry would be a Downplayed variant of that, and in fact he's already on that page's example list. But I could also see an argument for that being shoehorned and missing a Mook-level Sister Trope to Time-Limit Boss.
The main distinction being that when a Tonberry's time limit runs out, he starts beating the hell out of you. When a Time-Limit Boss's time limit runs out, it's over, full stop.
Hmm, I'll have to see about starting a YKTTW for it, if there isn't already an old one to dig up.
TV Tropes's No. 1 bread themed lesbian. she/her, fae/faerAlso related to Metal Slime, where the thing gets away (or does something else to take away a bonus for beating) it if you aren't fast.
Edited by Scorpion451

Do we have a trope for RPG enemies which have to be defeated in X number of turns before something bad happens, like they use a powerful attack (e.g. the Tonberry from the FF series, which edges closer and closer before using a powerful attack), summon more enemies or grow stronger if they aren't defeated?
Edited by Bisected8