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We enhance real-time search algorithms with bounded propagation of heuristic changes. When the heuristic of the current state is updated, this change is propagated consistently up to k states not necessarily distinct. Applying this idea to FALCONS, we have develop the new FALCONS(k), an algorithm that keeps the good theoretical properties of FALCONS and improves its performance. We provide experimental results on benchmarks for real-time search, showing the benefits of our approach.