Kraken: It breaks three cave columns, each one resulting in a wave, with the last sweeping you outside and it escaping.
Red dragon: As described, if you exit the large room (pretty much necessary eventually as it's really tough to battle it there) and can't kill it by the time the bridge ends, it falls down to its doom.
Chimera: Each floor of the tower gives way and you fight on each floor below. The last floor sends it to a lava lake, while you land on ground.
Killer Rabbit: Knights appear, carrying a crate of... you guessed it, Holy Hand Grenades. Well, regular bombs actually, but they do a ton of damage.
Medusa: You can flee the first room by exiting to the right, which leads to a further section of the stage, then a temple where you fight her again. Strangely enough, it seems that it cannot be killed in the temple, and no matter how fast you deplete her life, she retreats, so there's no point to fleeing in the first place as the fight isn't even easier.
Arch Demon: If you take too long, the female monk places the holy tablet and light rains from the heavens, damaging it.
Wraith: If you run out of candles, you'll be forced to keep going right until you exit the stage.
Cyclops: One of those that I'm not aware if there's any time limit or similar. I once tried to force it by killing each cyclops but leaving the hands alone so more cyclops would enter the fray, but after 6-7 of them I gave up. Anyone knows?
Gargoyle Gate: Same, no clue if there's a way to not kill it.