Metal fluorides, promising lithium-ion battery cathode materials, have been classified as conversion materials due to the reconstructive phase transitions widely presumed to occur upon lithiation. We challenge this view by studying FeF3 using X-ray total scattering and electron diffraction techniques that measure structure over multiple length scales coupled with density functional theory calculations, and by revisiting prior experimental studies of FeF2 and CuF2. Metal fluoride lithiation is instead dominated by diffusion-controlled displacement mechanisms, and a clear topological relationship between the metal fluoride F- sublattices and that of LiF is established. Initial lithiation of FeF3 forms FeF2 on the particle's surface, along with a cation-ordered and stacking-disordered phase, A-LixFeyF3, which is structurally related to α-/β-LiMn2+Fe3+F6 and which topotactically transforms to B- and then C-LixFeyF3, before forming LiF and Fe. Lithiation of FeF2 and CuF2 results in a buffer phase between FeF2/CuF2 and LiF. The resulting principles will aid future developments of a wider range of isomorphic metal fluorides.