Wolfgar said:
I believe they can appear and move, but they can't eat your face until your movement, right?
Yes. It happens during the mythos phase and you can only fight in the movement phase. 'A monster appears' encounters are an exception to that. Moving gates are also different - they suck (and delay) you in when they move on top of your investigator. I recently got lucky to avoid an opening gate through an encounter, which threw me out to the streets, only to have it chase me down. It was hilarious.