We have derived sets of pair potentials for the gamma-phases of CuCl and CuBr using a self-consistent approach. These potentials, together with that for gamma-CuI obtained previously, reproduce the sequential experimental trends in binding energies, Cu+ diffusion constants and the phonon density of states. Molecular dynamics simulations with these potentials have been used to explain the experimentally observed diffusion anomaly, that is the diffusion rate in CuI is similar to that in CuCl and lower than in CuBr. We conclude that this effect is due to a many-body interaction between dipoles induced on the large polarisable I- ions when Cu+ ’jump over’ the energy barrier between lattice and interstitial sites.