We present a new flow-based video completion algorithm. Previous flow completion methods are often unable to retain the sharpness of motion boundaries. Our method first extracts and completes motion edges, and then uses them to guide piecewise-smooth flow completion with sharp edges. Existing methods propagate colors among local flow connections between adjacent frames. However, not all missing regions in a video can be reached in this way because the motion boundaries form impenetrable barriers. Our method alleviates this problem by introducing non-local flow connections to temporally distant frames, enabling propagating video content over motion boundaries. We validate our approach on the DAVIS dataset. Both visual and quantitative results show that our method compares favorably against the state-of-the-art algorithms.