We consider a large-scale multi-robot path planning problem in a cluttered environment. Our approach achieves real-time replanning by dividing the workspace into cells and utilizing a hierarchical planner. Specifically, we propose novel multi-commodity flow-based high-level planners that route robots through cells with reduced congestion, along with an anytime low-level planner that computes collision-free paths for robots within each cell in parallel. A highlight of our method is a significant improvement in computation time. Specifically, we show empirical results of a 500-times speedup in computation time compared to the baseline multi-agent pathfinding approach on the environments we study. We account for the robot's embodiment and support non-stop execution with continuous replanning. We demonstrate the real-time performance of our algorithm with up to 142 robots in simulation, and a representative 32 physical Crazyflie nano-quadrotor experiment.