Unfortunately, some of them have reverted to their old ways and are up in the tree tops again. They're way out of reach for most players to battle. MOST players. ;)
If you're a night elf druid, I found a way to access these birds or any other out of reach wild pet.
1. Get into range of your target.Simple, yea? The trick is to not kill the pet, but to piss it off enough so that it will come towards you. Shadowmelding will take you out of combat, and if you're quick enough you can grab it for a battle right after. You may need to wait a moment for your cursor to turn back into a green paw print. Be careful not to right click on the pet if your cursor icon is still a sword!
2. Cast Faerie Fire onto it, which will put you and the pet into combat.
3. As soon as it's on top of you, shadowmeld and then quickly engage in a Pet Battle with it.
I haven't tried other class abilities that cause agro but don't do damage, but if there are other methods, any night elf could potentially use this trick to lure those pesky pets towards you.
A hunter or rogue of any class might also be able to do it since both have agro dumping abilities. The key is to find an attack that won't kill the pet.