Underwater Wireless Sensor Networks (UWSNs) are increasingly applied to explore underwater environments and resources. Some characteristics of mobile UWSNs, such as energy restriction, low transmission rate, node mobility and multiplicity, affect networks' performance significantly. In this paper, a Mobile-MultiSink (MMS) routing protocol is proposed for UWSNs with multiple mobile sink nodes. In the proposed MMS routing protocol, sensor nodes report events to multiple sink nodes using dynamic routing. There are two phases in this protocol, the layering phase and the construction phase. In the layering phase, each sink node establishes a layered structure, which needs to be refreshed periodically. Taking into consideration that the packet delivery rate (PDR) will reduce because of the mobility of nodes, we stipulate that every sink node should refresh its layered structure every time the PDR comes below a critical value. In the communication phase, the intermediate relay nodes are selected to forward data to the target sink node. This phase contains the selection of target sink node and the selection of relay nodes. We evaluate the performance of MMS using NS-3. Our findings demonstrate that the proposed protocol has better packet delivery rate and lower energy consumption in UWSNs with multiple mobile sink nodes.