Robotic Ultrasound Systems (RUSS) provide a promising solution to reduce operator dependency, alleviate physical strain, and meet the growing demand for ultrasound procedures. However, their clinical applicability remains limited by their inability to adapt to dynamic patient movements and tissue deformations during scans. This work introduces a novel framework that leverages Laplacian Trajectory Editing (LTE) for real-time adaptation of scan trajectories in response to both rigid and non-rigid patient movements. it integrates a RGB-D camera to capture surface point clouds, which are processed to estimate displacements between consecutive frames. These displacements define anchor points for LTE-based trajectory adaptations, ensuring smooth motion while preserving local trajectory properties. This approach is validated through experiments spanning rigid phantom movements, generalization across differently shaped phantoms, and non-rigid human arm motion. Adaptation accuracy is quantified by comparing adapted trajectories to a ground-truth reference, with root mean squared errors averaging 0.026 0.012 m in non-rigid scenarios. Real-time trajectory adaptation is achieved, with an average LTE adaptation processing time of 373 ms per trial. Furthermore, our implementation achieved low tracking errors across all conditions while maintaining a high success rate in diverse movement scenarios. These results demonstrate the feasibility of LTE for real-time trajectory adaptation in ultrasound scanning, offering a pathway to more autonomous and clinically viable RUSS implementations.