We consider moving points along a given path, with a fixed speed, so that no two points ever come closer than 1 (in the space into which the path is embedded, not only along the path) while they follow the path (all points traverse the path from start to finish). Since the motion of any point along the path is fully determined as soon as the point enters the path, our only decisions are the times when to send the points at the start of the path. We give algorithmic results for the problem of scheduling as many points as possible, i.e., maximizing the throughput.