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Short answer would be: It doesn't.

The local planner itself is run in a control loop by move_base, i.e., the local trajectory isn't passed off to another component to be executed. Only the velocity commands at the beginning of this trajectory are send to the robot (which must implement these within its controller). In the next timestep the trajectories are reevaluated given the current pose, sensor data, etc.

If nothing unexpected changed, the trajectory that was best before should still be the best in the next step and thus the velocity commands are send again. This closes the loop.