In the last few years, the proliferation of edge and cloud computing infrastructures as well as the increasing number of mobile devices has facilitated the emergence of many novel applications. However, that increase of complexities also creates novel challenges for service providers, for example, the efficient management of interdependent services during runtime. One strategy is to reallocate services dynamically by migrating them to suitable servers. However, not every microservice can be deployed as stateless instances, which leads to suboptimal performance of live migration techniques. In this work, we propose a novel live migration scheme focusing on stateful microservices in edge and cloud environments by utilizing the underlying messaging infrastructure to reconstruct the service's state. Not only can this approach be applied in various microservice deployment scenarios, experimental evaluation results also show a reduction of 19.92% downtime compared to the stop-and-copy migration method.