Dispersionless injections are a ubiquitous characteristic of substorms. They are defined as simultaneous enhancements in the fluxes of electrons and ions of different energies, and they are often observed near or inside geosynchronous orbit. We model dispersionless electron injections by considering the interaction of an earthward propagating electromagnetic pulse with the preexisting electron population. Such simulations have been performed previously <Li et al., 1993, 1998>; however, they assumed a constant propagation velocity for the transient fields. Observations have shown that substorm injections and associated magnetic signatures do not propagate at constant velocities, but rather slow down as they approach the inner magnetosphere. Between 4.5 and 6.6 RE the injection propagation speeds reach surprisingly low values, of the order of 24 km/s. Nonetheless, the injections still remain dispersionless <Reeves et al., 1996>. In our simulation we vary the pulse speed with the radial distance from the Earth to match the reported propagation speeds and demonstrate that dispersionless injections are achievable under such low propagation speeds. In particular, we simulate the dispersionless injections of 12 February 1991 measured at two radially displaced spacecraft (CRRES and LANL 1990--095), when they were both around local midnight.