Underwater Sensor Networks (USN) are used for harsh oceanographic missions where human operation is dangerous or impossible. Localization is essential for USNs. It is required for data tagging, node tracking and position-based routing algorithms. Localization is challenging because Global Positioning System (GPS) is not available in underwater; at the same time, existing GPS-less schemes based on fixed landmarks have high communication cost. Such cost is critical in Mobile Underwater Sensor Networks (MUSN), since sensor nodes drift with the ocean currents, thus requiring continuous refresh. In this paper, we propose a multi-stage localization scheme using mobile beacons. The beacons periodically ascent and descent in the water column. When they resurface, they receive new GPS coordinates. Then, they dive to the level of the underwater sensors to advertise these coordinates. In turn, localized sensors become proxy beacons and propagate their own coordinates, etc. This iterative, multi-stage localization is the major innovation of this paper. The goal is to localize the nodes with the smallest number of beacons using proxies instead, yet achieving an adequate accuracy. The major benefit is the reduction in operating costs. Mobility is a critical factor in determining performance. In this paper, performance (i.e., the percentage of localized nodes during a cycle, accuracy, delay and communication cost) is tested in a simulation scenario based on a realistic mobility model. The "Meandering Current Mobility with Surface Effect" (MCM-SE) model - a composite model combining surface and subsurface currents.