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This paper addresses the coupled navigation and control problem for fully actuated dynamical systems operating in cluttered environments. The proposed method composes local feedback control policies to address the global problem in a formally correct manner. Our contribution extends the method of sequential composition to a new class of flow-through policies, which are defined for an idealized point robot that is subject to bounds on velocity and acceleration. These policies, which are defined over convex polytopes, express natural behaviors and guarantee safe trajectories while respecting system constraints. This paper presents a constructive technique for building hybrid control systems with guaranteed transitions and stability properties. Simulation results demonstrate policy-based decision making for various initial conditions.