In this paper steady-state and transient diffusion and conduction in heterogeneous media, such as composite solids, are studied. The heterogeneities are of percolation type in the sense that a randomly-selected fraction of the system does not allow transport to take place. We present several new extensions of the classical effective-medium approximation (EMA) for diffusion and conduction in regular network models of heterogeneous media, including an EMA for topologically-random networks, such as the Voronoi network, for which we introduce a novel stochastic Green function, and a cluster (higher order) EMA for transient diffusion and other dynamical properties of heterogeneous media. In all cases the predictions are compared with the results of Monte Carlo simulations.