Open-loop end-to-end neural motion planners have recently been proposed to improve motion planning for robotic manipulators. These methods enable planning directly from sensor observations without relying on a privileged collision checker during planning. However, many existing methods generate only a single path for a given workspace across different runs, and do not leverage their open-loop structure for inference-time optimization. To address this limitation, we introduce Flow Motion Policy, an open-loop, end-to-end neural motion planner for robotic manipulators that leverages the stochastic generative formulation of flow matching methods to capture the inherent multi-modality of planning datasets. By modeling a distribution over feasible paths, Flow Motion Policy enables efficient inference-time best-of-$N$ sampling. The method generates multiple end-to-end candidate paths, evaluates their collision status after planning, and executes the first collision-free solution. We benchmark the Flow Motion Policy against representative sampling-based and neural motion planning methods. Evaluation results demonstrate that Flow Motion Policy improves planning success and efficiency, highlighting the effectiveness of stochastic generative policies for end-to-end motion planning and inference-time optimization. Experimental evaluation videos are available via this \href{https://zh.engr.tamu.edu/wp-content/uploads/sites/310/2026/03/FMP-Website.mp4}{link}.