Based on the development of networking services and the emergence of various computing environments, such as edge and cloud environments, appropriate service allocation based on processing delays, computational resources, data volumes, and other requirements has become crucial. To support dynamic load shifting and reduce service downtime, service allocation should be performed using live migration, which maintains network consistency. There have been many studies on live migration techniques and their implementation in software applications. However, limited research has been reported on field-programmable gate array (FPGA)-based transparent service migration. This paper proposes and evaluates a transparent live-migration architecture for an FPGA computing system that maintains network consistency before and after migration and enables effective migration without any software or hardware extensions on communicating edges or cloud nodes. The proposed architecture achieves sufficient throughput and sufficiently low downtime to be introduced into cloud and edge nodes.