Another way to do it might be to take a leaf out of the Heroquesting rules and frame it as a sequence of contests. Maybe dividing by distinct objectives (find out where they're going, get transport, overcome obstacles, actually catch them etc.) or just dividing the chase into stages.
Losing a stage would get the usual impairments, winning would get a bonus (there are "Lingering Benefits" rules which could be a bit fiddly if used in full but might be rather better in a sequence of contests).
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