In mobile sensor network localization problems we seek to estimate the position of the mobile sensor nodes by using a subset of pair-wise range measurements (among the nodes and with mobile anchors). When the sensor nodes are static, convex relaxations have been shown to provide a remarkably accurate approximate solution to this NP-hard estimation problem. In this paper, we propose a novel convex relaxation to tackle the more challenging dynamic case and we develop a moving horizon convex estimator based on a maximum a posteriori (MAP) formulation. The resulting estimator is then compared to standard extended and unscented Kalman filters both with respect to computational complexity and performance with simulated data. The results are promising, yet a more detailed analysis is needed.