We consider theoretically the transport of a point-sized Brownian particle in a two-dimensional channel with periodically varying cross-section. When the channel width changes slowly, the transport process is associated with the concept of an “entropy barrier,” where the change in the number of available “states” for the Brownian particle governs the transport process. Using generalized Taylor-Aris dispersion theory and long-wavelength asymptotics, we determine the mean particle velocity and effective diffusivity (dispersivity) for two cases: electrophoretic transport in an insulating channel and motion under the influence of a constant force. At the same time, we arrive at rational definitions for the concept of an entropy barrier as a function of the driving force. For the case of electrophoresis, we find that the “incompressibility” of the electrophoretic velocity field implies that the concept of an entropy barrier is satisfied by the geometric criterion dw/dx << 1, where w(x) is the channel width and x is the longitudinal coordinate along the direction of net transport. The dispersivity evolves smoothly between hindered transport at low Peclet numbers to enhanced diffusivity at large Peclet numbers, owing to an effective “electrical lubrication” along the channel walls caused by their impenetrability to electric current. For the case of a constant force of magnitude F, the entropy barrier concept requires the dynamical criterion dw/dx << Pe^-1/2, where Pe is the Peclet number FL/kT for a channel period of 2L and kT is the Boltzmann factor. The latter restriction, which arises from an analysis of the transport equation in the long-wavelength limit, is compared to Brownian dynamics simulations. In contrast to the electrophoretic case, the dispersivity in the presence of a constant force exhibits a maximum at moderate Pe. We compare our asymptotic results for large Pe to simulation results.