Multi-robot motion planning problems often have many local minima. It is essential to visualize those local minima such that we can better understand, debug and interact with multi-robot systems. Towards this goal, we present the multi-robot motion explorer, an algorithm which extends previous results on multilevel Morse theory by introducing a component-based framework, where we reduce multi-robot configuration spaces by reducing each robots component space using fiber bundles. Our algorithm exploits this component structure to search for and visualize local minima. A user of the algorithm can specify a multilevel abstraction and an optimization algorithm. We use this information to incrementally build a local minima tree for a given problem. We demonstrate this algorithm on several multi-robot systems of up to 20 degrees of freedom.