The relentless growth of mobile Internet of Things (IoT) devices has shifted computation toward a distributed computing continuum, spanning edge, fog, and cloud layers, where energy efficiency, low latency, and dynamic node mobility are critical yet often conflicting goals. Existing scheduling frameworks struggle to balance these demands under real-world conditions, especially as device movement and heterogeneous workloads increase system complexity. We present MOSAIC, a mobility-aware scheduling and resource management framework designed to optimize performance in dynamic IoT environments. Our approach introduces three key innovations. First, a refined five-tier architecture extends the traditional edge-fog-cloud hierarchy by adding proximity, local, and regional mobility layers, enabling computation to follow mobile users more effectively and reducing unnecessary network traffic. Second, MOSAIC integrates a preemption-aware dynamic scheduler with an Adaptive-$\lambda$reinforcement learning-based resource manager that adapts based on workload changes and mobility patterns, prioritizing energy-efficient edge execution while meeting strict deadlines. Third, the framework utilizes real-world mobility traces, including Levy-Walk, Random-Walk, and Geolife, to drive reconfiguration and improve decision accuracy. We evaluate MOSAIC through a large-scale deployment across three geographically distributed regions of the Grid'5000 testbed, using realistic workflows and mixed periodic/DAG task loads. Our results show that, compared to state-of-the-art schedulers, MOSAIC reduces energy consumption by 35.9%–×1.5, lowers latency by 42.8%–×4.9, and shortens makespan by 22.6%–×7.2, all while maintaining 100% deadline satisfaction across diverse mobility scenarios.