The object of research is the processes of dynamic allocation of computational tasks in heterogeneous Edge-Fog infrastructures of remote patient monitoring systems. The problem of dynamic distribution of computational tasks in remote health monitoring systems was addressed. In particular, ensuring a balance between medical data processing latency, energy consumption of wearable devices and compliance with deadlines. Additionally, user mobility, changes in the quality of wireless communication and the cost of transferring tasks between Edge and Fog nodes were taken into account. Data confidentiality was ensured using secure communication channels, remote attestation and a trusted environment for execution. A system model of dynamic distribution of RPM tasks in a multi-level Cloud-Fog-Edge-IoT infrastructure was developed. The model covers ECG classification (Electrocardiogram, ECG), arrhythmia detection, temperature analysis and user activity recognition. An online offloading algorithm based on Lyapunov theory (Migration-Gated Lyapunov Offloading, MGLO) is proposed. The algorithm accounts for energy consumption, delay, deadlines and the costs of changing the execution node. To protect medical data, the use of TEE (Trusted Execution Environment), secure communication channels and remote attestation of Fog nodes is provided. To validate the solutions, a series of simulations were conducted in the YAFS (Yet Another Fog Simulator) environment with a comparison of the Edge only, LP and Horizon MILP scenarios in modes with and without handover support. It was experimentally established that the proposed MGLO algorithm provides effective distribution of RPM tasks in a mobile Edge-Fog environment. In the mode without handover support, the total energy consumption of wearable devices decreased by more than 2 times compared to the Edge only scenario (from 1,354.74 mWh to 630.15 mWh). In the handover mode, the energy consumption decreased by more than 5 times (from 1,354.74 mWh to 259.39 mWh). The proportion of deadline violations was 0.14% and 0.12%, respectively. The average decision time was 1.9 ms (54–85 times faster compared to the Horizon MILP scenario). Overall, the results confirmed the feasibility of using MGLO for online offloading management in remote health monitoring systems.