In this paper we derive the equations governing the transport of energy and mass in a porous medium saturated by a multiphase multi-constituent fluid mixture under conditions that yield steep continuous moving fronts and abrupt discontinuous moving phase-change interfaces. In these equations the volume fractions of each phase, the potentials, such as, the temperature, pressure and concentration, and the corresponding fluxes, are permitted to jump in value across the phase-change interfaces. As an example of application of the derived continuum model, we specialize the theory to the problem of propagation of melting/freezing interfaces in a salt-water saturated porous medium for a coupled process governed by heat flow, fluid flow and species transport.