When it comes to real-time applications, where response time is paramount, fog computing is an excellent method for improving cloud computing performance. The fog computing paradigm effectively reduces latency and enhances throughput by positioning cloud resources in proximity to Terminal Devices (TDs). The task offloading (TO) from Terminal Devices was thoroughly investigated; however, the challenge of TD mobility has not received adequate consideration, which is the focus of the current study. In this context, TD movement in fog computing pertains to the service transfer, when the terminal device moves through one fog into another one, necessitating meticulous coordination among fogs. A system is presented to facilitate collaboration between fog computing and the cloud, primarily to continuously monitor the present location of the assigned offloading TD and to effectively execute the job in a dispersed manner whereas the TD is in motion. A structure designates two queues within each fog: one for incoming tasks that arrive TDs and another for transfer tasks from various fogs, utilizing a robust inter-fog communications mechanism to ensure all relevant components are informed of their current situation. A Python software is being developed to replicate a framework and exemplary operational scenarios. A program has been utilized to conduct comprehensive tests to evaluate the framework's performance under varying mobility settings. The results demonstrate that the framework is very dependable and can provide the appropriate reaction to the correct TD at the optimal time across diverse mobility modalities.