This paper exploits sink mobility to prolong the network lifetime in wireless sensor networks where the information delay caused by moving the sink should be bounded. Due to the combinational complexity of this problem, most previous proposals focus on heuristics and provable optimal algorithms remain unknown. In this paper, we build a unified framework for analyzing this joint sink mobility, routing, delay, and so on. We discuss the induced subproblems and present efficient solutions for them. Then, we generalize these solutions and propose a polynomial-time optimal algorithm for the origin problem. In simulations, we show the benefits of involving a mobile sink and the impact of network parameters (e.g., the number of sensors, the delay bound, etc.) on the network lifetime. Furthermore, we study the effects of different trajectories of the sink and provide important insights for designing mobility schemes in real-world mobile WNNs.