Cloud Data Centers have become the backbone infrastructure to provide services. With the emerging edge computing paradigm, computation and networking capabilities have been pushed from clouds to the edge to provide computation, intelligence, networking management with low end-to-end latency. Service migration across different computing nodes in edge and cloud computing becomes essential to guarantee the quality of service in the dynamic environment. Many studies have been conducted on the dynamic resource management involving migrating Virtual Machines to achieve various objectives, such as load balancing, consolidation, performance, energy-saving, and disaster recovery. Some have investigated to improve and predict the performance of single live migration of VM and container. Recently, several studies service migration in the edge-centric computing paradigms. However, there is a lack of surveys to focus on the live migration management in edge and cloud computing environments. We examine the characteristics of each field and conduct a migration management-centric taxonomy and survey. We also identify the gap and research opportunities to guarantee the performance of resource management with live migrations.