We give a simple picture of transient and stationary transport in lipid nanotubes connecting two vesicles, when a difference of membrane tension is imposed at time t= 0, either by pressing one vesicle with a micro-fiber, or by adding a surplus of membrane lipid. The net result is a transport of membrane from the tense towards the floppy vesicle. In the early stage, the tube remains cylindrical, and the gradient of surface tension gives rise to two opposite flows of the internal liquid: a Marangoni flow towards the direction of high tension, and a Poiseuille flow (induced by Laplace pressures) in the opposite direction. At longer time, the tube reaches a stationary state, where curvature and Laplace pressure are balanced. Marangoni flows dominate for giant vesicles, where Laplace pressure is negligible.