Honestly, the way I've seen it play out in most stories I've read is that the monster's initial evolution is tied to survival mechanics in the new world. They eat weird magical plants, absorb core energy from fallen foes, or accidentally trigger a mutation just by existing. In 'So I'm a Spider, So What?', the System literally forces it with skills and evolution paths, which feels a bit like a LitRPG. But then there's this weird second phase where the evolution becomes psychological. A slime learns to mimic human emotion, a goblin king starts questioning the hierarchy it was born into. That's where it gets messy, because the character has to reconcile its monstrous instincts with whatever morality it's picking up from the heroes or villains around it. I've dropped a few series where that internal conflict just got hand-waved.
Sometimes the world-hopping itself becomes the catalyst for change. Jumping from a high-magic world to a tech-heavy one forces a magical creature to adapt in a completely different way, like developing a resistance to iron or learning to disguise itself as machinery. That's a fun twist, but it's rarer. Most authors just keep piling on bigger horns and more tentacles without really changing the core being, which gets repetitive fast. The best ones make you forget they're a monster halfway through, until a moment of crisis reminds everyone—including the reader—of what's lurking underneath the surface.