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Maneuverability under stress or in difficult conditions. Flying isn't about the inherent ability to fly, it's about how well you fly when using a spell or other sort of device.
When it has to do with conditions, this is what I'd generally call an internal game, or a game within a game. When it has to to with combat tactics, it's got a direct application to an existing set of rules. When it's a game within a game, you're talking about a randomized way of applying something from the character sheet to a way of getting a benefit or suffering a setback.
Tests like this are either one-or-the-other (fall or stay mounted on flying carpet, accurately get between the wind-whipped pillars, etc), or there might be degrees of success (how fast you reach the objective 10 miles away through the wind).
In either case, the skill of flying pretty much applies only to magic-users, since that's the only class with any possible training, but there has to be a way of determining it for members of other classes as well.
Method 1 (simple resolution)
For one-or-the-other, binary-type tests
For non-magic-users, use a DEX check if it's about close-in flying. Use a WIS check if it's about long-distance flying through obstacles. Give magic-users a bonus on the roll (if this makes sense under the circumstances). Alternatively, use a saving throw (this focuses on level rather than attributes). The more doctrinaire you are about being "old school" the more you lean toward character level as the most important factor. I think in this situation that would be dogmatic, but opinions may vary.
Method 2 (series of challenges)
If the situation is an important one, and is more like a game within a game, it's either just extended random checks (sketchy in terms of testing player skill, but exciting) or it's something where player skill comes into play (more interesting but can also bog down the pace). Player skill requires a puzzle or a tactical decision based on knowledge, and I can't off the top of my head think of how flying would give rise to this kind of a game. For a pure excitement-boost series of randomized checks, the situation would be based on a series of changing conditions (wind, obstacles, etc). Possibly even these are randomly determined.
Given that player skill and character level have no real bearing on a flying-type challenge, I would tend to avoid these, and treat flying success as either completely automatic, or determined by a base chance if it's distance travel. It can be used as an excitement booster, but if that's the case, make it short and fairly easy. Excitement boosters should ALWAYS have a second chance to pull the fat out of the fire. If for no other reason than to boost the tension even higher, but also to avoid the feeling that a bad outcome was determined entirely by the dice.
Art: The Flying Carpet by Viktor Vasnetsov (1880)
Music: Of course.
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