The paper addresses the problem of transport of inert solutes in random nonstationary velocity fields. General equations for the first two moments of the particle trajectory are derived. It is shown that they can be solved analytically in quadratures for the case of quasi-unidirectional mean flows. The solution generalizes the previous results obtained for transport in velocity fields which are uniform in the average. The theory is applied to the case of transport in media displaying a linear trend in the mean logconductivity. The dynamics of particle displacements is discussed for the case of two-dimensional media with isotropic fluctuations of the logconductivity. ¿ American Geophysical Union 1996