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Offloading computationally intensive tasks from mobile users (MUs) to a virtualized environment such as containers on a nearby edge server, can significantly reduce processing time and hence end-to-end (E2E) delay. However, when users are mobile, such containers need to be migrated to other edge servers located closer to the MUs to keep the E2E delay low. Meanwhile, the mobility of MUs necessitates handover among base stations in order to keep the wireless connections between MUs and base stations uninterrupted. In this paper, we address the joint problem of container migration and base-station handover by proposing a coordinated migration-handover mechanism, with the objective of achieving low E2E delay and minimizing service interruption. The mechanism determines the optimal destinations and time for migration and handover in a coordinated manner, along with a delta checkpoint technique that we propose. We implement a testbed edge computing system with our proposed coordinated migration-handover mechanism, and evaluate the performance using real-world applications implemented with Docker container (an industry-standard). The results demonstrate that our mechanism achieves 30%-40% lower service downtime and 13%-22% lower E2E delay as compared to other mechanisms. Our work is instrumental in offering smooth user experience in mobile edge computing.