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Recent dynamical analyses suggest that some Jupiter family comets (JFCs) may originate in the main asteroid belt instead of the outer solar system. This possibility is particularly interesting given evidence that icy main-belt objects are known to be present in the Themis asteroid family. We report results from dynamical analyses specifically investigating the possibility that icy Themis family members could contribute to the observed population of JFCs. Numerical integrations show that such dynamical evolution is indeed possible via a combination of eccentricity excitation apparently driven by the nearby 2:1 mean-motion resonance with Jupiter, gravitational interactions with planets other than Jupiter, and the Yarkovsky effect. We estimate that, at any given time, there may be tens of objects from the Themis family on JFC-like orbits with the potential to mimic active JFCs from the outer solar system, although not all, or even any, may necessarily be observably active. We find that dynamically evolved Themis family objects on JFC-like orbits have semimajor axes between 3.15 au and 3.40 au for the vast majority of their time on such orbits, consistent with the strong role that the 2:1 mean-motion resonance with Jupiter likely plays in their dynamical evolution. We conclude that a contribution from the Themis family to the active JFC population is plausible, although further work is needed to better characterize this contribution.