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We present a dynamical study of 39 active Centaurs and 17 high-perihelion (q> 4.5 au) Jupiter-family comets (JFCs) with a focus on investigating recent orbital changes as potential triggers for comet-like activity. We have identified a common feature in the recent dynamical histories of all active Centaurs and JFCs in our sample that is not present in the history of the majority of inactive population members: a sharp decrease in semimajor axis and eccentricity occurring within the past several hundred years prior to observed activity. We define these rapid orbital changes as “a-jumps.” Our results indicate that these orbital reshaping events lead to shorter orbital periods and subsequently greater average per-orbit heating of Centaur nuclei. We suggest that thea-jumps could therefore be a major trigger of cometary activity on Centaurs and JFCs. Our results further imply that analyses of the recent dynamical histories could be used to identify objects that are currently active or may become active soon, where we have identified three such Centaurs with recenta-jumps that should be considered high-priority targets for observational monitoring to search for activity.