Traditionally, the Ogston regime is studied solely in the limit of low field intensities. This explains why the theoretical discussion has focused until now on the relative roles of the fractional volume available to the analyte and the subtleties of the gel architecture. Over the past several years, we have developed a lattice model of gel electrophoresis that has allowed us to revisit the fundamental assumptions of the standard Ogston model. In particular, we demonstrated that the fractional free volume is not the relevant parameter for gel sieving. In this article, we continue the development of this model and we generalize our mathematical approach to treat nonvanishing electric field intensities. To do so, we must revisit the way biased random walks are normally modeled by stochastic processes. Straightforward generalizations based on standard Metropolis-like schemes fail at high field intensities. Moreover, our generalization requires the complete decoupling of the spatial directions parallel and perpendicular to the field direction. We show that our novel theoretical approach makes it possible to calculate exact mobilities in the presence of lattice obstacles. Several two-dimensional examples are then studied, including one that includes topological dead ends that act like traps. In the latter case, we recover results very similar to those reported by Serwer [Biopolymers 29, 1863 (1990)] on the trapping electrophoresis of charged spheres in agarose gels. In the absence of such traps, the mobility is shown to be a very weak function of the electric field, thus validating the historical neglect of the field intensity in the development of obstruction models for the Ogston sieving regime of small analytes. Finally, we describe how the present model could be improved to treat more realistic cases and we discuss the problem of the field dependence of the diffusion coefficient during electrophoresis.