We investigate the electrokinetic flow and transport within a micro-electrophoresis device. A mathematical model is set up, which allows to perform two-dimensional, time-dependent finite-element simulations. The model reflects the dominant features of the system, namely electroosmosis, electrophoresis, externally-applied electrical potentials, and equilibrium chemistry. For the solution of the model equations we rely on numerical simulations of the core region, while the immediate wall region is treated analytically at leading order. This avoids extreme refinements of the numerical grid within the EDL. An asymptotic matching of both solutions and subsequent superposition, nevertheless, provides an approximation for the solution in the entire domain. The results of the simulations are verified against experimental observation and show good agreement.