Evidence for cometary activity beyond Jupiter's and Saturn's orbits-such as that observed for Centaurs and long-period comets-suggests that the thermal processing of comet nuclei starts long before they enter the inner solar system, where they are typically observed and monitored. Such observations raise questions as to the depth of unprocessed material and whether the activity of Jupiter-family comets (JFCs) can be representative of any primitive material. Here we model the coupled thermal and dynamical evolution of JFCs, from the moment they leave their outer solar system reservoirs until their ejection into interstellar space. We apply a thermal evolution model to a sample of simulated JFCs obtained from dynamical simulations that successfully reproduce the orbital distribution of observed JFCs. We show that due to the stochastic nature of comet trajectories toward the inner solar system, all simulated JFCs undergo multiple heating episodes resulting in significant modifications of their initial volatile contents. A statistical analysis constrains the extent of such processing. We suggest that primordial condensed hypervolatile ices should be entirely lost from the layers that contribute to cometary activity observed today. Our results demonstrate that understanding the orbital (and thus, heating) history of JFCs is essential when putting observations in a broader context.