In this paper, we study moving-target search, where an agent (=hunter) has to catch a moving target (=prey). The agent does not necessarily know the terrain initially but can observe it within a certain sensor range around itself. It uses the strategy to always move on a shortest presumed unblocked path toward the target, which is a reasonable strategy for computer-controlled characters in video games. We study how the agent can find such paths faster by exploiting the fact that it performs A^* searches repeatedly. To this end, we extend Adaptive A^*, an incremental heuristic search method, to moving-target search and demonstrate experimentally that the resulting MT-Adaptive A^* is faster than isolated A^* searches and, in many situations, also D^* Lite, a state-of-the-art incremental heuristic search method. In particular, it is faster than D^* Lite by about one order of magnitude for moving-target search in known and initially unknown mazes if both search methods use the same informed heuristics.