Fog computing is a viable approach to improving the performance of cloud computing, especially in terms of response time, which is critical to real-time applications. Specifically, the fog brings the cloud resources closer to terminal devices (TDs), thereby decreasing latency and increasing throughput. The problem of task offloading from TDs to the fog has enjoyed much research work, but the issue of TD mobility has not found enough attention, and hence is the present work. Herein, TD mobility in fog computing involves the transfer of services while the TD is moving from one fog to another, requiring delicate coordination between the fogs. To this end, a framework is proposed to ensure that the fogs together with the cloud collaborate, first to always keep track of the current location of the TD offloading the task, and second to accurately serve the task in a distributed fashion while the TD is moving. The framework dedicates two queues in each fog, one to receive fresh tasks from TDs and one to receive hand-over tasks from other fogs, and leverages a vigilant inter-fog messaging system capable of keeping all concerned components abreast of the latest status. A program has been written in Python to simulate the framework and example operational environments. The program has been used to perform extensive experiments in order to assess the performance of the framework under high and low mobility conditions. The findings indicate that the framework is highly reliable and can deliver, under various mobility modes, the right response to the right TD at the right time.