The emergence of fog computing has significantly enhanced real-time data processing by bringing computation resources closer to data sources. This adoption is very beneficial in the healthcare sector, where abundant time-sensitive processing tasks exist. Although such adoption is very promising, there is a challenge with the limited computational capacity of fog nodes. This challenge becomes even more critical when mobile IoT nodes enter the network, potentially increasing the network load. To address this challenge, this paper presents a framework that leverages a Many-to-One offloading (M2One) policy designed for modelling the dynamic nature and time-critical aspect of processing tasks in the healthcare domain. The framework benefits the multi-tier structure of the fog layer, making efficient use of the computing capacity of mobile fog nodes to enhance the overall computing capability of the fog network. Moreover, this framework accounts for mobile IoT nodes that generate an unpredictable volume of tasks at unpredictable intervals. Under the proposed policy, a first-tier fog node, called the coordinator fog node, efficiently manages all requests offloaded by the IoT nodes and allocates them to the fog nodes. It considers factors like the limited energy in the mobile nodes, the communication channel status, and low-latency demands to distribute requests among fog nodes and meet the stringent latency requirements of healthcare applications. Through extensive simulations in a healthcare scenario, the policy’s effectiveness showed an improvement of approximately 30% in average delay compared to cloud computing and a significant reduction in network usage.