The standard Mythos phase has 4 steps:
1. Open Gate and Spawn Monster(s)
2. Place Clue Token(s)
3. Move Monsters
4. Activate Mythos Ability
Kingsport adds two steps (no official names in the rules):
a. Move rifts
b. Advance rift progress and open rift if full
Obviously, rifts will open after they move (first sentence under "Open Rifts", p. 8), but do they move before or after monsters move? If before, the monsters they generate may also move; if after, the new monsters won't move.
Also, do they move before or after the Mythos' ability is activated? If before, generated monsters may be removed by the ability.
