Hi,
It seems that Operation: Relocation is broken.
You are supposed to collect 8x ‘Wild Proto Avian’ but there doesn’t appear to be any in the area. Only one player can tag any mob, and they spawn incredibly slow.
This reminds me of when Spill the Tea quest came out - only one player could tag any of the mobs, and you eventually fixed it.
Please do not design quests in which only one player can tag a mob at a time. This is bad design.
The Wild Proto Avians are friendly NPCs spawning throughout the zone, and if you happen to have the effects turned on they glow. You just need to approach them and use the hand, then they will lasso, and you bring them to the broker next to the animal trap.
You can also collect multiple and turn them in at once, even mounted. This part of the quest doesn’t require combat. Hope that helps! 
(PS I was stuck for a bit too. I didn’t realize where the broker turn-in was so my birds kept despawning.)