Recent measurements show that magnetospheric convection electric fields during the main phases of magnetic storms are much more complicated in spatial structure than the electric fields that have generally been used to model formation of the stormtime ring current. To investigate the transport effects of such more realistic stormtime electric fields on magnetospheric charged particles, we map the Assimilative Model of Ionospheric Electrodynamics (AMIE) electric potential functions analytically with a simple magnetic field model at selected times of interest during several magnetic storms from 1997--1998 and during the extremely large storm of July 2000. We calculated corresponding contours of constant Hamiltonian (kinetic plus potential energy) for first invariant values that range from 0 to 30 MeV/G (representative of equatorially mirroring cold plasma, ring current, and radiation-belt ions and electrons). These equatorial contours would constitute drift paths if the electric field were truly constant in time. We thus calculated how far along such quasi-drift paths the corresponding particles would have drifted after specific amounts of time, and we compare these quasi-drift characteristics with those obtained from a simple semiempirical model of the convection electric field. We find considerable variability among stormtime equatorial quasi-drift paths, reflecting the known variability of AMIE equipotentials. Patterns of equatorial quasi-drift in the simplified electric field model are (by construction) symmetric about the dawn-dusk meridian. During the main phases of storms, the equatorial electric field derived from AMIE tends to be strongest in an MLT sector several hours wide on the night side. This concentration of AMIE equipotentials provides a channel for rapid transport (requiring ~20--30 min) for ions with first invariant values representative of the ring current population from the nightside neutral line to low L values (~3--4) near dusk, where the partial ring current forms. During the extremely large Bastille Day storm of 15 July 2000 (minimum Dst = -300 nT) the drift patterns derived from AMIE show penetration of ions to as low as L ~ 2. This deep penetration of ring current ions could help to account for the very strong ring current that was observed during this storm. The quasi-steady state drift properties help us anticipate the implications of a more realistic electric field model for the particle drifts that lead to the formation of the stormtime ring current.