In this battle, you have to seize Arete's throne to the NW corner of the map. Like the previous maps you've battled in Valla, most of them are separated into floating islands so having a considerable number of reliable fliers in your army will help a lot. If not, you can use your underleveled fliers as transports by partnering them up with your powerful ground units and switching them to engage enemies. There are also three chests, all in the western side of the map. None of the enemies carry keys so you'll have to bring in your characters with the Locktouch skill to open them.
The floating square boxes will move from two fixed locations after every turn but you have to place your ground units there in order to move around. This is very inconvenient and restrictive. However, if you have no choice, it's the only other way to move your ground forces. There are no dragon veins that will modify the paths of your ground units so this is the only way to move around the map.
One of the enemy fliers is “Scarlet”. Start by moving east ward, towards the chests. Keep an eye of the enemy fliers' movements since they can reach your forces at the start of the second turn. It's better to let them come to you so can position your units defensively and that your light units don't get taken out during the fliers' turn. Clear the nearby enemies in the first island to the west then ready your units to take out the enemies near the chests.
You can bring your most powerful units in the middle of the enemies, including powerhouses like your avatar, Ryoma, and/or Xander, especially if they're supported by their best partners. Once the enemies are clear near the treasure chests, bring in your lockpickers and loot their contents.
Chest contents:
SW: Chakram
W: Ragnarok, 10000g
Once done, make your way to the Queen's position and defeat her. Seize the throne to complete the mission.