What do you think this should represent? Can you make a "Perform Stunt" to "run fast", Athletics (running) check and get more than one manoeuvre from your sacrificed action? Overall I would like bonus movement from Perform Stunt to be more cinematic (grabbing ropes and swinging etc.) but that's not always an option.

Bonus white to run-based checks normally. Allow 2nd specialization to gain "swift".

What's a normal run check? Would that me something like spending extra manoeuvers to run across tricky terrain and the check being any required because you were moving fast and facing hazards (woods in my game, where doing so requires check to avoid falling over something).

If you were staging a race or a chase as an encounter in and of itself, that might use one or more Athletics or Coordination checks for "normal" running. You wouldn't want to use the standard abstract movement rules for that sort of thing, because there'd be no random element and dwarves would always win every foot-race despite their shorter legs, just because they had the Toughness needed to suck up Fatigue. Which sounds terribly broken, until you realize it's no more bizarre than D&D where every unarmored human runs at the same 40" pace. In either game, if a plot point hinged on a race or chase, the GM would be wise to ignore the movement-in-combat rules and call for some sort of attribute or skill check instead.

Chases and races are relatively rare, so there's not a lot of value to taking a running specialization. It's pretty much a wasted specialization in most campaigns using the rules as written. That's ok, though, as specializations aren't really meant to be huge. They're just the door prize you get for training a skill in a career you intend to complete. Even if you roll a particular specialization all the time, it's still just a tiny drop in any given die pool. The best specialization is still much weaker than just about anything else you could spend an XP on. If a player was going to "waste" something on their character sheet for the sake of flavor and character development, a specialization slot is a pretty small sacrifice to make.

I have toyed with the idea of letting the PCs, or perhaps anyone who fulfill the right requirements, take an athletics check when moving between ranges you need several manueovres to traverse to reduce the fatigue you suffer.

Bonus white to run-based checks normally. Allow 2nd specialization to gain "swift".

That should only apply for movement, though - you get one aditional manoeuvre that can be spent to move further than you normally could without any penalty, but not for managing equipment and suchlike. I think that might require a similar specialisation in coordination.