Interconnected systems have to route information to function properly: At the lowest scale neural cells exchange electrochemical signals to communicate, while at larger scales animals and humans move between distinct spatial patches and machines exchange information via the Internet through communication protocols. Nontrivial patterns emerge from the analysis of information flows, which are not captured either by broadcasting, such as in random walks, or by geodesic routing, such as shortest paths. In fact, alternative models between those extreme protocols are still eluding us. Here we propose a class of stochastic processes, based on biased random walks, where agents are driven by a physical potential pervading the underlying network topology. By considering a generalized Coulomb dependence on the distance on destination(s), we show that it is possible to interpolate between random walk and geodesic routing in a simple and effective way. We demonstrate that it is not possible to find a one-size-fit-all solution to efficient navigation and that network heterogeneity or modularity has measurable effects. We illustrate how our framework can describe the movements of animals and humans, capturing with a stylized model some measurable features of the latter. From a methodological perspective, our potential-driven random walks open the doors to a broad spectrum of analytical tools, ranging from random-walk centralities to geometry induced by potential-driven network processes.