If your searching for the legendary birds, here is something you should know:
Spoiler:It appears the legendary birds are dependent on your starter; Articuno if you picked Chespin, Zapdos if you picked Fennekin and Moltres if you picked Froakie.
If you manage to get the legendary birds, which are roaming, to go to Azure Bay, then they will settle in Sea Spirit's Den and finally be battleable
Hope that helps. :)
Does this mean that you can only catch one out of the three legendary birds?