Membrane nanotubes (NTs) and their networks play an important role in intracellular membrane transport and intercellular communications. The transport characteristics of the NT lumen resemble those of conventional solid-state nanopores. However, unlike the rigid pores, the soft membrane wall of the NT can be deformed by forces driving the transport through the NT lumen. This intrinsic coupling between the NT geometry and transport properties remains poorly explored. Using synchronized fluorescence microscopy and conductance measurements, we revealed that the NT shape was changed by both electric and hydrostatic forces driving the ionic and solute fluxes through the NT lumen. Far from the shape instability, the strength of the force effect is determined by the lateral membrane tension and is scaled with membrane elasticity so that the NT can be operated as a linear elastic sensor. Near shape instabilities, the transport forces triggered large-scale shape transformations, both stochastic and periodic. The periodic oscillations were coupled to a vesicle passage along the NT axis, resembling peristaltic transport. The oscillations were parametrically controlled by the electric field, making NT a highly nonlinear nanofluidic circuitry element with biological and technological implications.