Healthcare monitoring is crucial, especially for the daily care of elderly individuals living alone. It can detect dangerous occurrences, such as falls, and provide timely alerts to save lives. Non-invasive millimeter wave (mmWave) radar-based healthcare monitoring systems using advanced human activity recognition (HAR) models have recently gained significant attention. However, they encounter challenges in handling sparse point clouds, achieving real-time continuous classification, and coping with limited monitoring ranges when statically mounted. To overcome these limitations, we propose RobHAR, a movable robot-mounted mmWave radar system with lightweight deep neural networks for real-time monitoring of human activities. Specifically, we first propose a sparse point cloud-based global embedding to learn the features of point clouds using the light-PointNet (LPN) backbone. Then, we learn the temporal pattern with a bidirectional lightweight LSTM model (BiLiLSTM). In addition, we implement a transition optimization strategy, integrating the Hidden Markov Model (HMM) with Connectionist Temporal Classification (CTC) to improve the accuracy and robustness of the continuous HAR. Our experiments on three datasets indicate that our method significantly outperforms the previous studies in both discrete and continuous HAR tasks. Finally, we deploy our system on a movable robot-mounted edge computing platform, achieving flexible healthcare monitoring in real-world scenarios.