If there is a time lapse between chapters, the narrator should be able to allow players to discard goals and resources.
Since I cannot move characters from one story to a new story, then sequels must be in the same line. So there should be a way to dump leftover cards that no longer apply, from the last story.
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Elfwreck commented
I think there are a few posts with this basic idea. I expect that, for now, the solution is for the narrator to post an "epilogue" scene with a number of challenges to match the remaining cards that need to be eliminated (or more than one epilogue scen, if some characters need to drop more than 3 cards), in which they can reminisce over the adventure and drop cards as they go.
So: Epilogue; everyone meets in the tavern and has a drink to celebrate that they're all still alive. (6 challenge points: Strong result - you all get roaring drunk had have a fantastic time. Weak result - same as strong, but you're going to have hangovers in the morning.) For more challenges, add the bartender -- and spend cards telling him about your terrific exploits.
Adjust as appropriate to venue. Murder mysteries recap the story in a drawing room; medical dramas explain to a review board what happened; deep-space adventures on a single ship discuss how they'd cope with similar problems differently in the future. The epilogue could even be "each person records recent events in their journals."
This is not an optimal solution, but it would allow dropping cards without creating an entirely fake scene that just says "generic challenge: 5 points" and having players throw their cards onto it.