Electrophoresis, the motion of charged species through liquids and pores under the influence of an external electric field, has been the principle source of chemical pumping for numerous micro- and nanofluidic device platforms. Recent measurements of ion currents through single or few carbon nanotube channels have yielded values of ion mobility that range from close to the bulk mobility to values that are two to seven orders of magnitude higher than the bulk mobility. However, these experiments cannot directly measure ion flux. Experiments on membranes that contain a large number of nanotube pores allow the ion current and ion flux to be measured independently. Here, we report that the mobilities of ions within such membranes are approximately three times higher than the bulk mobility. Moreover, the induced electro-osmotic velocities are four orders of magnitude faster than those measured in conventional porous materials. We also show that a nanotube membrane can function as a rectifying diode due to ionic steric effects within the nanotubes.