A theoretical model of electromigrative, diffusive and convective transport in polymer-gel composites is presented. Bulk properties are derived from the standard electrokinetic model with an impenetrable charged sphere embedded in an electrolyte-saturated Brinkman medium. Because the microstructure can be carefully controlled, these materials are promising candidates for enhanced gel-electrophoresis, chemical sensing, drug delivery, and microfluidic pumping technologies. The methodology provides solutions for situations where perturbations from equilibrium are induced by gradients of electrostatic potential, concentration and pressure. While the volume fraction of the inclusions should be small, Maxwell's well-known theory of conduction suggests that the model may also be accurate at moderate volume fractions. In this work, the theory is used to compute ion fluxes, electrical current density, and convective flow driven by an electric field applied to an homogeneous composite. The electric-field-induced (electro-osmotic) flow is a sensitive indicator of the inclusion $\zeta$-potential and size, electrolyte concentration, and Darcy permeability of the gel, while the electrical conductivity is usually independent of the polymer gel and is relatively insensitive to characteristics of the inclusions and electrolyte.