With nothing to do but wait for news, you spend some time with Y'shtola and Alisaie.
This quest is available if you have completed the quest Under the Moonlight.
Starting the Quest
Talk to Alisaie in the Rising Stones in Mor Dhona to begin the quest.
Claim the Nearby Table
Go to the quest icon on the map and check on the sparkling Destination. There will be several cutscenes in sequence. Say Yes if you are ready.
Solo Duty: Emissary of the Dawn
Your hotbar/crossbar is replaced with a small number of actions. Ruin is a single-target attack. Physick restores the target's HP. Tri-shackle is a multi-target attack. Starstorm is a high-damage multi-target attack. Sort Pet Hotbar allows you to change the order of these actions.
Go south and you find some Airship Wreckage. Use Ruin on it until it is destroyed. You can also auto-attack it to do a small amount of additional damage.
After it is destroyed, use Physick on the Wounded Popularis. After that, check on the sparkling Wounded Popularis to proceed.
Stand near the Moonstone Carbuncle so that you can get the Shining Moonstone status from standing in the blue circle. It will boost the damage that you deal and reduce the damage that you take. Use Tri-Shackle on the Hostile Strangers and Hostile Commander, but switch to attacking with Ruin when there are only two enemies left.
After you defeat them, there will be a cutscene.
Afterward, stand in the blue circle around the carbuncle while using Ruin on the enemies that are attacking you and Maxima. When more enemies appear, use Tri-Shackle on them. WHen Hostile Sky Armors chain the Menacing Mercenary, use Starstorm (the Limit Break) on them all. Defeat the remaining enemies and there will be a cutscene.
Speak with Alisaie
Talk to Alisaie to complete the quest. You will learn the
Next Main Scenario Quest
After you complete the quest, talk to Alisaie to accept the next quest: Sisterly Act. There will be a cutscene.