The article presents studies on the Taylor transport in disordered systems. We assume that a moving particle can exist in different states and that for each state its transport properties are described by a different linear evolution operator. The transition from one state to another is described by a Markov process in continuous time. The transition rates of the Markov process depend on a set of variable parameters that are randomly selected from a known probability density. This type of transport model is of interest in connection with the study of multiphasic transport, for example, in the case of chromatographic separation, neutron migration in nuclear reactions, or neutrino flow in astrophysical problems. We consider two different types of effective medium approximations: (1) a global approach, which consists, of solving the Taylor problem for a given set of values of the of the random parameters, followed by averaging the result, and (2) a local approach. for which the averaging is done for small macroscopic regions of the system, which leads to a system of age-dependent master equations. We show that the averaging in the global approach induces an artificial collective behavior that results in ballistic diffusion. The local approach may lead either to normal or dispersive transport. We apply our theory to the experiment of Drazer and Zanette (Phys. Rev. E. 1999, 60, 5858) and show that that the local approach is in agreement with experimental data.