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.X.H. is supported by funding from EPSRC Doctoral Prize, Adolphe Merkle and the Swiss National Science Foundation (Program NRP70 No. 153990) and European Commission via MSCA (Grant 798169). A.S.E. acknowledges financial support from the Royal Society. E.C.M. acknowledges funding from European Commission via MSCA (Grant 747449) and RTI2018-094550-A-100 from MICINN. Z. L. acknowledges funding from the Faraday Institution via the FutureCat consortium. C.J.P. is supported by the Royal Society through a Royal Society Wolfson Research Merit award, and EPSRC grant EP/P022596/1. A.L.G. acknowledges funding from the ERC (Grant 788144). This research was supported as part of the North Eastern Center for Chemical Energy Storage, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, and Office of Basic Energy Sciences under Award Number DE-SC0001294. Work done at Argonne and use of the Advanced Photon Source, an Office of Science User Facility operated for the US Department of Energy (DOE) Office of Science by Argonne National Laboratory, was supported by the US DOE under Contract No. DE-AC02-06CH11357. Work done at Diamond Light Source was under Proposal EE17315-1. The authors are grateful to Prof. G. Ceder and other NECCES members for many stimulating discussions concerning fluoride-based conversion reactions and on the origins of structural hysteresis. The authors also acknowledge the help from S. Dutton, T. Dean, A. Docker, M. Leskes and D. Keeble