Stochastic thermostats commonly used in molecular dynamics trajectories are known, under certain conditions, to exhibit a synchronization effect whereby trajectories initialized at different points in phase space synchronize to a single master trajectory if they are subjected to the same sequence of random forces. We investigate the spatiotemporal robustness of this effect analytically and with molecular dynamics simulations in one and three dimensions in the strong coupling limit. We first investigate the response of the system to a time- and spacewise local perturbation and show that desynchronization behaves diffusively at long times for infinite systems. We then explore the behavior of temporally persistent but spatially local perturbations and observe strikingly different behaviors as a function of dimensionality: in one dimension, the desynchronization propagates through the whole lattice and grows with time, while in three dimensions, the desynchronization remains localized in the neighborhood of the perturbation.