Lipid vesicles can be connected by membrane nanotubes to build networks with promising bioanalytical properties. Here we characterize electrophoretic transport in such membrane tubes, with a particular eye to how their soft-material nature influences the intratube migration. In the absence of field, the tube radius is 110 +/- 26 nm, and it remains in this range during electrophoresis even though the applied electric field causes a slight decrease in the tube radius (similar to 6-11%). The electrophoretic velocity of the membrane wall (labeled with quantum dots) varies linearly with the field strength. Intratube migration is studied with latex spheres of radii 15, 50, 100, and 250 nm. The largest particle size does not enter the tube at fields strengths lower than 1250 V/m because the energy cost for expanding the tube around the particles is too high. The smaller particles migrate with essentially the same velocity as the membrane at low fields. Above 250 V/cm, the 15 nm particles exhibit an upward deviation from linear behavior and in fact migrate faster than in free solution whereas the 100 nm particles deviate downward. We propose that these nonlinear effects arise because of lipid adsorption to the particles (dominating for 15 nm particles) and a pistonlike compression of the solvent in front of the particles (dominating for 100 nm). As expected from such complexities, existing theories for a sphere migrating in a rigid-wall cylinder cannot explain our velocity results in lipid nanotubes.