Fluorite-related materials are encountered in nuclear applications as both fuel materials and for the immobilization of minor actinides. Many of these compounds display great resistance to radiation-induced amorphization, but there seem to be differences in the detailed mechanism. We compare and contrast the observed behavior induced by ion irradiation versus the behavior at high temperatures of some systems with generic compositions close to δ-A4B3O12, γ-A2B5O13, and β-A2B7O17 where A is a rare-earth element like Scandium and B is a tetravalent transition metal. In these structures, cation disorder and anion order compete and constitute an interesting playground for studying the separate behaviors of the two sublattices. Acknowledgment: This work was supported in part by the French Grant ANR-10-LABX-0039-PALM.