This paper deals with the problem of allocating goals to multiple robots with complex dynamics while computing obstacle-free motions to reach those goals. The spectrum of existing methods ranges from complete and optimal approaches with poor scalability, to highly scalable methods which make unrealistic assumptions on the robots and/or environment. We overcome these limitations by using an efficient graph-based method for decomposing the problem into sub-problems. In particular, we reduce the problem to a Minimum-Cost Max-Flow problem whose solution is used by a multi-robot motion planner that does not impose restrictive assumptions on robot kinodynamics or on the environment. We show empirically that our approach scales to tens of robots in environments composed of hundreds of polygons.