To Run Away you have to roll the die. You escape on a 5 or more.
Some Class and Race abilities and some Treasures make it easier or harder to Run Away from all monsters.
Some monsters give you a bonus or penalty to your roll for that monster only.
If you are fleeing from multiple monsters, you roll separately to escape each one, in any order you choose, and suffer Bad Stuff from each one that catches you as soon as it catches you.
You (and anyone helping you in Combat) roll separately for each monster, in whatever order you like. (If you can’t decide, roll the die; the high roller decides who Runs Away first.) The monster(s) CAN catch you both.
Once you have resolved all Run Away rolls, discard the monster(s).