Is Poyaqui moving to Lions Arch?
No? When you use the map and it circles the port in LA for him it’s actually telling you to take the ship there to Zephyr Sanctum to face him (just in case you got the mail from one of the other instructors without going to Zephyr Sanctum and therefore had idea how to get there -_-). I’m not entirely sure how they will deal with it after the event, it is possible the new map will stay open and just the ship will leave?
Considering the achievement specifically states that you must defeat him during the event and that he holds an unlock to a permanent mini-game, he’ll probably be relocated once zephyr sanctum leaves. Where he’ll be located next, you’ll just have to wait and see.