This paper presents a technique for dynamically reconfiguring search spaces in order to enable Bayesian autonomous search and tracking missions with moving targets. In particular, marine search and rescue scenarios are considered, highlighting the need for space reconfiguration in situations where moving targets are involved. The proposed technique improves the search space configuration by maintaining the validity of the recursive Bayesian estimation. The advantage of the technique is that autonomous search and tracking can be performed indefinitely, without loss of information. Numerical results first show the effectiveness of the technique with a single search vehicle and a single moving target. The efficacy of the approach for coordinated autonomous search and tracking is shown through simulation, incorporating multiple search vehicles and multiple targets. The examples also highlight the added benefit to human mission planners resulting from the technique's simplification of the search space allocation task.