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We compare the performance of various dynamic query abolishment mechanisms in different unstructured overlay network topologies such as found in several P2P systems. We specifically focus on techniques based on iterative deepening and checking. Both unintelligent and intelligent variants of the methods are used in the study. Additionally, we propose a new mechanism called the chasing wave based on the use of increasing delays for search packets on the forwarding nodes. We show that the proposed chasing wave algorithm trades effectively the increase in propagation delay to substantially lower overhead. The performance of the methods are compared in several network configurations and using several metrics. We make concrete proposals on the suitability of the specific dynamic query abolishment methods for different search algorithms.