How long can creatures fly when being used as mounts?

The DMG listing is slightly wrong, mathematically.

At 80'/round, you travel 48,000 feet an hour. Divide by 5,280' for mph, and that ends up being just a smidge over 9 miles an hour.

So it is 27 miles, rest, 27 miles, rest, 27 miles, done. So that is 81 miles a day, not 72.

And if you, like me, believe that number to be ridiculously slow to stay airborne for such a creature, you'd be right. I'm doubling their airspeed, as long as they are not in any combat, just traveling, to 18 miles an hour.

And that gives them 54 miles, rest, 54 miles, rest, 54 miles, final rest or 162 miles in 12 hours or 13.5 miles an hour on average in a 12 hour day.

Good enough.

The DMG addresses this on p. 119, under the section titled "The Sky":

Flying by spell or magic item works the same as travel on foot, as described in the Player’s Handbook. A creature that serves as a flying mount must rest 1 hour for every 3 hours it flies, and it can’t fly for more than 9 hours per day. Thus, characters mounted on griffons (which have a flying speed of 80 feet) can travel at 8 miles per hour, covering 72 miles over 9 hours with two 1-hour-long rests over the course of the day. Mounts that don’t tire (such as a flying construct) aren’t subject to this limitation.

As stated: Mounted flying creatures can fly up to 9 hours a day, plus two intervening 1-hour rests.