Here's a tip for the leves: Look at what the target is for your leve. Leves that have you tracking down and exterminating certain targets are likely to flee. For example, the Ul'dah leve "Revenge of the Dodos" has you killing just Dodos, there's no disguises or item hunting involved. This level may result in a target trying to make a break for it so as a rule of thumb, take down the last two targets at the same time.
Leves where you have to uncover a hidden enemy (Firestarter Imps or Carrion Ladybugs for example.) won't have you chasing down a target, so far I've yet to have one instance where a target fled when doing these. Another is if you're gathering items (Floatstone, Cactuar Leg, etc.) from a series of enemies they won't attempt to flee. As soon as the last item is obtained, the remaining targets will despawn. Quite honestly most leve targets fall pretty quickly (They can hit hard, have high defense, but usually are much weaker in terms of HP than their normal brethren.) so unless you lack ranged attack skills, you can often kill them before they reach their destination and spawn more.
If an enemy is moving too fast, sheath your weapon and start running, you move much faster in cases like this and the enemy normally stops for a second or two when it's fleeing giving you time to catch up. Once you're close enough, unleash a ranged attack or spell. Fortunately unlike FFXI, you can keep casting as you move. Start up a spell and keep running, it will still go off without fail. If you don't kill the target, sheath your weapon and repeat the process again.
FINAL FANTASY XIV Roleplayer
Sair Gammonari - Hyur Midlander Male - Conjurer (Somewhat retired.)
Mihana Zhralyia - Miqo'te Seeker of the Sun Female - Archer