The conformational dynamics of DNA molecules undergoing electrophoresis on a fluid substrate-supported cationic lipid bilayer is investigated using fluorescence microscopy. At low electrophoretic velocities, drift of 2-D random coils is observed. In contrast, at velocities larger than 0.3 μm/s, the DNA molecules stretch out and assume branched configurations. The cross-over scenario is explained by the observation that cationic lipids segregate underneath the adsorbed DNA and confine the DNA to its counter charge imprint on time scales shorter than the relaxation time of the imprint. The concept of a tube-like confinement of the DNA is corroborated by the observed 1/N size dependence of the electrophoretic mobility in analogy to the biased reptation model in gels. The role of membrane defects and possible applications of membrane-based electrophoresis in microfluidic devices are discussed.