[en] In the quantum theory of diffusion of light interstitials (such as the positive muons or hydrogen atoms) in metals the Condon approximation commonly is used, e.g., the effects of phonon fluctuation on th tunneling integrals are neglected. Here results beyond this approach are reviewed concerning coherent tunneling, incoherent multi-phonon transitions, and incoherent one-and two-phonon processes. With respect to the latter processes some new results are presented which indicate that beyond the Condon approximation the rate of incoherent muon transitions at low temperature may increase linearly with temperature. (orig.)