We study turbulent transport of passive tracers by random wave fields of a rather general nature. A formalism allowing for spatial inhomogeneity and anisotropy of an underlying velocity field (such as that caused by a latitudinally varying Coriolis parameter) is developed, with the aim of treating problems of large-scale ocean transport by long internal waves. For the special case of surface gravity waves on deep water, our results agree with the earlier theory of Herterich & Hasselmann (1982), though even in that case we discover additional, off-diagonal elements of the diffusion tensor emerging in the presence of a mean drift. An advective diffusion equation including all components of the diffusion tensor D plus a mean, Stokes-type drift u is derived and applied to the case of baroclinic inertia–gravity (BIG) waves. This application is of particular interest for ocean circulation and climate modelling, as the mean drift, according to our estimates, is comparable to ocean interior currents. Furthermore, while on the largest (100 km and greater) scales, wave-induced diffusion is found to be generally small compared to classical eddy-induced diffusion, the two become comparable on scales below 10 km. These scales are near the present limit on the spatial resolution of eddy-resolving ocean numerical models. Since we find that uz and Dzz vanish identically, net vertical transport is absent in wave systems of this type. However, for anisotropic wave spectra the diffusion tensor can have non-zero off-diagonal vertical elements, Dxz and Dyz, and it is shown that their presence leads to non-positive definiteness of D, and a negative diffusion constant is found along a particular principal axis. However, the simultaneous presence of a depth-dependent mean horizontal drift u(z) eliminates any potential unphysical behaviour.