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In surveillance problems, the uncertainty in the position of a target can be specified in terms of a probability distribution. To reduce the average search times to detect a target using mobile sensors, it is desirable to have the trajectories of the sensors sample the probability distribution uniformly. When the target is moving, the initial uncertainty in the position of the target evolves forward in time according to the target dynamics. We assume a model for the dynamics of the target and build upon our previous work for stationary targets to define appropriate metrics for uniform coverage of the evolving probability distribution. Using these metrics, we derive centralized feedback control laws for the motion of the sensors so that they achieve uniform coverage of the moving target distribution. We demonstrate the performance of the algorithm on various scenarios.