We study the phenomenon of internal avalanching within the context of recently introduced lattice models of granular media. The avalanche is produced by pulling out a grain at the base of the packing and then studying the number of grains to be rearranged before the packing is stable again. We find that the avalanches are long-ranged, decaying as a power law. We study the distribution of avalanches as a function of the density of the packing and find that the avalanche distribution is a very sensitive structural probe of the system.