In wireless sensor networks, the existing real-time routing protocols for stationary sinks exploit a spatiotemporal approach, utilizing the delivery speed based on the end-to-end distance to fulfill the desired time deadline. This approach cannot be directly applied to a mobile sink since the distance can be varied depending on its movement. That is, the delivery speed cannot be determined without knowing the intermediate sensors to know the current location update of the mobile sink and control their transmission speeds. We focus on the real-time protocol for slowly varying mobile sink compared to transmission speed, which may be the most common case. By the slowly varying constraint, the movable area of a sink can be determined by the two factors: its moving distance and initial location. The movable area could provide the maximum transmission distance between a source and a mobile sink. This letter proposes a real-time data dissemination protocol for mobile sinks utilizing the movable area concept. Simulation results show the proposed protocol provides high success ratio of real-time requirement.