We're getting into the really nitty-gritty now, but something about rules crunching gives me the good feelings, so....

Per RAW in 5e, a character's average marching time is 3 miles an hour, or 300 feet per minute. I couldn't find any hard rules about equations for characters slower or faster than your average 30 speed, but it appears to be a simple equation: 1/10 of your speed in miles per hour, 10 times your speed in feet per minute. This checks out if you consider a round to be ten seconds - obviously, ten rounds at thirty feet per round is three hundred feet. The rules appear to also account for moving at a faster pace, with that speed being moved up to 4mph/400fpm; more than normal marching speed, but significantly less than if we were to do stuff like dash or roguish actions. That makes a lot of sense, since I'm not confident even a level 20 fighter could maintain a dead sprint for more than a even a couple minutes at most, just like you said.

That said, the the way the travel speed lines up with the average movement speed of 30 indicates to me that magically or otherwise "passive" assistance to movement would result in a to-scale faster travel time, though with obviously massive penalties to perception and the like. If anything, the fact that characters can march at speeds faster than mathematically possible on a combat map makes me think that standard combat movement speed isn't actually the average character's top speed.

With that in mind, I'd say that anything that allows you to move faster without conscious effort (an action, concentration, etc.) would linearly affect world travel speed, even when the movement per round starts getting up into the hundreds.