This paper addresses the problem of coordinating multiple robots in a common environment with positive velocity along fixed paths. We propose a priority-based framework, which is an overlay of the coordination space approach that fits well the positive velocity constraint. The theoretical developments rely on a priority graph that defines the relative order of the robots. As a primary contribution, the priority graph is proved to uniquely encode the homotopy classes of feasible trajectories, and a mathematical characterization of priority cycles leading to deadlock configurations is provided. As a secondary contribution, the framework is applied to the problem of designing an intersection management system. This system uses heuristics rules to assign priorities between vehicles, and once the the priorities are fixed the algorithm is proved to possess optimality properties. Good performance is observed in numerical experiments on an open network of intersections. More generally, the framework seems to open up new avenues of research in the field of coordination.