In chronic pain rehabilitation, physiotherapists adapt physical activity to patients' performance based on their expression of protective behavior, gradually exposing them to feared but harmless and essential everyday activities. As rehabilitation moves outside the clinic, technology should automatically detect such behavior as to provide similar personalized support. Previous works have shown the feasibility of automatic Protective Behavior Detection (PBD) within a specific predefined activity at the time. In this paper, we investigate the use of deep learning to detect protective behavior across activity types, using wearable motion capture and surface electromyography data collected from healthy participants and people with chronic pain. We approach the problem by continuously detecting protective behavior within an activity rather than estimating its overall presence. The best performance for our activity-independent system reaches mean F1 score of 0.82 with Leave-One-Subject-Out validation. Performances remain competitive when protective behavior is modelled separately per activity type (mean F1 score: bend-down=0.77, one-leg-stand=0.81, sit-to-stand=0.72, stand-to-sit=0.83, reach-forward=0.67). Such performance reaches excellent level of agreement with the average experts' rating, suggesting clear potential for personalized chronic-pain management at home. We analyze various parameters characterizing our approach to better understand how the results could generalize to other PBD datasets and different groundtruth granularity.