The magnetic dipole interaction mechanism is shown to be important in a nonradiative energy transfer process of a lanthanide ion in a centrosymmetric environment. The concentration quenching of near-ultraviolet luminescence in Cs,NaTmCI, may be explained by this mechanism, the calculated nonradiative transfer rate from the donor Tm'+ being 1.44~ IO5 s-'. The transfer rate decreases by a factor of IO3 for transfer to the 7th nearest neighbours, compared with that to the 1st nearest. However, the angular dependence of the transfer, and the different coordination numbers for nth nearest neighbours, lead to an increased transfer rate to 3rd nearest neighbours, compared with that to the 2nd nearest set: and to the 5th nearest neighbours compared with the 4th set. The dipole-dipole magnetostatic transfer mechanism is shown to account for 'D,-'F,, energy migration between Eu'+ ions in Cs*NaY, _XEuXC16. However, many other cross-relaxation processes of Ln 3+ in elpasolite lattices appear to take place by nonresonant energy transfer, and these processes should be described by a coupling of the donor and acceptor vibronic states.