In the first two papers of this series, a model, cosistent with observations, was proposed for the densification of a ceramic powder matrix in the presence of dense inclusions. One aspect of the model concerned the breakup of grain bridges linking polycrystalline clusters. This breakup phenomenon results in the formation of large cracklike voids. The mechanism that produces this breakup phenomenon, which we calleddesintering, is the subject of this third paper, where we (1) present further experimental results concerning the breakup of polycrystalline bridges that undergo grain growth and (2) calculate the free energy of two evolving bridge configurations. The instability of grain bridges is shown to be a function of the grain size to bridge length ratio. That is, when the ratio exceeds a critical value, the bridge will break up, e.g., desinter, to reduce the free energy of the system. This instability condition can be independently achieved either by grain coarsening at constant bridge length or by extending the bridge length under an applied tensile stress at constant grain size.