Roughly 1000 white dwarfs are known to be polluted with planetary material, and the progenitors of this material are typically assumed to be asteroids. The dynamical architectures which perturb asteroids into white dwarfs are still unknown, but may be crucially dependent on moons liberated from parent planets during post-main-sequence gravitational scattering. Here, we trace the fate of these exomoons, and show that they more easily achieve deep radial incursions towards the white dwarf than do scattered planets. Consequently, moons are likely to play a significant role in white dwarf pollution, and in some cases may be the progenitors of the pollution itself.