The perception of 3D motion of surrounding traffic participants is crucial for driving safety. While existing works primarily focus on general large motions, we contend that the instantaneous detection and quantification of subtle motions is equally important as they indicate the nuances in driving behavior that may be safety critical, such as behaviors near a stop sign of parking positions. We delve into this under-explored task, examining its unique challenges and developing our solution, accompanied by a carefully designed benchmark. Specifically, due to the lack of correspondences between consecutive frames of sparse Lidar point clouds, static objects might appear to be moving – the so-called swimming effect. This intertwines with the true object motion, thereby posing ambiguity in accurate estimation, especially for subtle motion. To address this, we propose to leverage local occupancy completion of object point clouds to densify the shape cue, and mitigate the impact of swimming artifacts. The occupancy completion is learned in an end-to-end fashion together with the detection of moving objects and the estimation of their motion, instantaneously as soon as objects start to move. Extensive experiments demonstrate superior performance compared to standard 3D motion estimation approaches, particularly highlighting our method's specialized treatment of subtle motion.