You specifically fire the moves only when you’re in that mode of play. It’s super specific that way.
I’m not sure if family play will react to character moments or if character play is there to carry out family goals. Probably it’s supposed to be both, right? Like, you could probably just do one or the other. I could see using family play as a campaign generator type thing, like using Mighty Empires to set up Warhammer battles. That seems obvious. But maybe if you really wanted the game to be more character-driven, then family play becomes downtime play in support of what the characters are up to.
My gut tells me, mostly based on the existence of the Quick Character rules (little half-sheet playbooks you toss together if there’s no reason for focus characters to be together), that Jay Iles doesn’t really want the game to be super character-driven. It’s a small-ish worry I have, tbh, because pretty much all my players heavily prefer character-driven play. So it might feel weird or unsatisfying to them to play the worldbuilding game at the expense of the character experience.
I also feel like the paucity of character moves points at deprioritizing character play. The game feels quite purposeful in its design decisions. So far I trust that they’re there for good reasons.