How can I Axtinguish enemy Pyros?

Since the flamethrower projectiles have a travel time, you have to time it so that by the time the flame hits the pyro, and before it goes away due to afterburn immunity, you have your axtinguisher out already hitting the pyro.

Here's an example by sketchek, a popular TF2 youtuber:

In the video you'll notice that from a high distance (which is much easier to get critical axtinguisher kills on enemy pyro), he jumps down, shoots the flamethrower down. Since it has travel time, the pyro is travelling at roughly the same speed as the flames. By the time he's nearly all the way down, he has axtinguisher out and ready to melee the pyro. When he hits the bottom, the flames have hit the Pyro but not triggered afterburn immunity, allowing you to crit the pyro.

This doesn't have to be done while airborned, as you can see from the second video you can critflare on pyros who are being airblasted upwards if you time it correctly.