Previous studies indicate that more than a quarter of all white dwarf (WD) atmospheres are polluted by remnant planetary material, with some WDs being observed to accrete the mass of Pluto in 10^6 years. The short sinking timescale for the pollutants indicate that the material must be frequently replenished. Moons may contribute decisively to this pollution process if they are liberated from their parent planets during the post-main-sequence evolution of the planetary systems. Here, we demonstrate that gravitational scattering events among planets in WD systems easily triggers moon ejection. Repeated close encounters within tenths of a planetary Hill radii are highly destructive to even the most massive, close-in moons. Consequently, scattering increases both the frequency of perturbing agents in WD systems, as well as the available mass of polluting material in those systems, thereby enhancing opportunities for collision and fragmentation and providing more dynamical pathways for smaller bodies to reach the WD. Moreover, during intense scattering, planets themselves have pericenters with respect to the WD of only a fraction of an AU, causing extreme Hill-sphere contraction, and the liberation of moons into WD-grazing orbits. Many of our results are directly applicable to exomoons orbiting planets around main sequence stars.