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In a wireless sensor network with mobile sinks, the maintenance of efficient data delivery structure often brings a large amount of control overhead, which may offset the benefit of introducing mobile sinks. In this paper, we propose a multi-stage data routing protocol (called MLRP) to address this problem. To reduce the protocol overhead, MLRP integrates layered Voronoi scoping and dynamic anchor selection. Such design is expected to greatly reduce the diffusion scope of sink movement updates and thus reduce the frequency at which the data delivery structure is refreshed. Simulation results show that MLRP can effectively reduce the protocol overhead while ensuring high packet delivery ratio as compared with existing work.