Talents or expenditures of AP that allow a PC to move and do something or perform a special move do not refresh Per Arena Talents, correct?
This is the assumption I have been going on for the longest time. It is also my preference... especially with the core rules containing Charger and my own homebrew classes containing many Talents and Inherent Abilities that facilitate free/extra movement.
(Has this been asked yet and I just haven't seen it? If so, my apologies)
I'm not sure. I think they might.
The important thing is that per-arena Talents that move you between Arenas don't refresh themselves. They're refreshed once you've moved after you've used them.
How you move after is never specified, so it may not have to be specifically using the Move action. Being Thrown or Pushed might refresh them as well.
Ultimately I'd leave it up to individual DM decision, but that's probably how I'd rule if asked on the spot.
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