Multi-robot collision-free and deadlock-free navigation in cluttered environments with static and dynamic obstacles is a fundamental problem for many applications. We introduce MRNAV, a framework for planning and control to effectively navigate in such environments. Our design utilizes short, medium, and long horizon decision making modules with qualitatively different properties, and defines the responsibilities of them. The decision making modules complement each other and provide the effective navigation capability. MRNAV is the first hierarchical approach combining these three levels of decision making modules and explicitly defining their responsibilities. We implement our design for simulated multi-quadrotor flight. In our evaluations, we show that all three modules are required for effective navigation in diverse situations. We show the long-term executability of our approach in an eight hour long wall time (six hour long simulation time) uninterrupted simulation without collisions or deadlocks.