How to get the DLC weapons in Fire Emblem Fortune's Weave

Fire Emblem: Fortune's Weave's DLC weapons give your army a sizeable advantage, though they come with a few drawbacks. Getting them requires completing a new subquest, and each of the four playable Flame Lords gets a special weapon. There's some timeline funkiness involved in carrying the weapons into Part 2 or beyond if you've already completed someone's Part 1.