Au deposition on Si(110) at temperature higher than 633 K leads to the formation of elongated liquid nano-droplets. In this work we show that under an electric eld, the droplets move along the [1-10] axis towards the positive electrode by electromigration. During the motion, they dissolve the substrate forming a hole underneath and a nanowire behind them. We develop a model highlighting that the driving force of these processes is bound to the respect (i) of the Au-Si bulk chemical equilibrium and (ii) of the correct balance of surface/interfacial energies. The motion takes place by means of sudden jumps with length of the order of a few µm and by small readjustments of the order of some nm of the droplet front and back edges. With respect to experiments without electric eld, electromigration changes the droplet motion by widening the droplet dissolution front and by facilitating droplet jumps.