Monstrous kaliri are huge owls bred by the arakkoa. These huge avians are bred within the bounds of Skettis itself, and are often seen patrolling the skies above the city. The Sha'tari Skyguard regularly sends daring fliers to bomb the monstrous kaliri eggs before they can hatch, thus preventing the possibility of losing any more of their numbers to them.
You can avoid their attack by flying up and right or left at the same time as soon as they aggro on you, if you can do so before you're dismounted (or if you're a druid), they end up looping up next to you as you move upward and are unable to attack.
Another method is to fly up and straight until reaching maximum altitude, then going into a steep dive. Usually, by the time you go into the dive, you'll be at the edge of their chase range. This works because even rare flying mounts can climb faster than the kaliri.
In case you actually want to hunt these creatures (e.g. for the daily cooking quest , you will need to draw their aggro. You can not attack them from the ground as they fly too high, and can not attack anything while flying, so the only option is to fly close enough to one to make it attack you. It's handy to be over water while doing so, in case the attack dismounts you. When your level gets to be higher than the kaliri's the aggro radius becomes quite small, and drawing the aggro can become difficult.
Patch 2.4.0 (25-Mar-2008): Monstrous Kaliri have learned how to chase players going straight up.
The Monstrous Kaliri are one of the most hated mobs in all of Outland by players. When this beast first appeared, players got seriously frustrated by them, causing many protests of "NERF MONSTROUS KALIRI" or "Get rid of Kaliri". People still hate them to this day.
However, since Patch 3.3.0 replaced being dazed off your mount with the "knock-back effect" they pose a smaller threat.