Brownian dynamics simulations are used to characterize the time scales involved in polymer electrophoresis through narrow constrictions. The polymer is modeled as a freely jointed bead-rod chain with a total charge distributed uniformly among the beads. The narrow constriction is a thin channel with height h(s) < R-g which separates two thicker channels, both of height h(1) similar to R-g where R-g is the polymer radius of gyration. The polymer is initially placed in a thick channel, and an applied electric field drives it into the next thick channel through the intervening narrow constriction. We find that the electrophoresis of the polymer is characterized by three time scales, each of which depends on the polymer chain length, N. An approach time, tau(app), describes the motion of the polymer to the entrance of the thin channel. Upon reaching the entrance of the thin channel, the polymer is entropically trapped, and its escape from the trap is associated with an activation time, T., After the activation event, the motion of the polymer through the thin channel and into the next thick channel is characterized by a crossing time, tau(cross). We find that whereas tau(app) and tau(act) decrease with N, tau(cross) increases with N. As a consequence, it is found that the transit velocity of the polymer, v(transit), first increases with N and then decreases beyond a certain value of N. The position of the maximum In v(transit) is shown to depend on the applied electric field strength, the relative values of h(s) and h(1), and whether the channel is two-dimensional or three-dimensional. We discuss the relevance of this behavior to polymer electrophoresis in microfluidic channels exhibiting entropic trapping effects and polymer translocation through nanopores.