| quindraco |
No. Climb, Swim, Fly, and Burrow by and large prevent exotic movement (there are exceptions, but by exotic movement, I mean any movement that isn't simply taking a move action to move - I don't think any of them permit Guarded Step, for example). Most of the rules you need are on pages 247 and 259.
For running:
Burrow and Climb explicitly ban the run action on page 259, and Swimming explicitly allows it. Flying does not mention it, so we have to use other rules text. The critical piece is the first line of the Flying block: "If you have a fly speed, you can use move actions to fly through the air." There is no rule in the book letting a creature with a fly speed take a full action to fly, and while you could argue that they ought to have it unless told they can't, the same logic would apply to creatures without a fly speed, as there's also nothing in the run action description on page 248 banning anybody from running in the air like Wile E. Coyote. So, it seems like you would only be able to fly-run if given a rule letting you, and there is no such rule, so you can't.
ShieldLawrence
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If a rule references speed without specifying a movement type, it refers to whatever movement type you are using.
The section on flying has inclusive rules about what you can do with move actions, and has nothing against using full actions. Therefore, we can use the above clause to determine that we could use full actions like run while flying.
Similarly, this allows Operatives with other movement speeds to actually use those movement types during Trick Attack. (You need the climb speed itself though, using athletics to climb still won’t work.)