In this article, the problem of the relay pursuit–evasion game is investigated. The game involves multiple pursuers and a single evader, which are all subject to the constraint of bounded curvature. To solve this problem, we develop a novel relay pursuit scheme that allows the single evader to be captured by the pursuers. First, the round pursuit strategy is derived as an alternative strategy to the circle straight pursuit strategy to further unlock the capture potential of the static pursuers. Next, the planar space is partitioned into some separate dominance regions by comparing the capture times required for each pursuit strategy. In consequence, a relay pursuit scheme based on the dominance region is obtained. At the end, a set of experiment results are presented to reveal the practicality and superiority of the approach developed in this article.