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Source-Free Domain Adaptive Object Detection transfers knowledge from a labeled source domain to an unlabeled target domain while preserving data privacy by restricting access to source data during adaptation. Existing approaches predominantly leverage the Mean Teacher framework for self-training in the target domain. The exponential moving average (EMA) mechanism in the Mean Teacher stabilizes the training by averaging the student weights over training steps. However, in domain adaptation, its inherent lag in responding to emerging knowledge can hinder the rapid adaptation of the student to target-domain shifts. To address this challenge, Dual-rate Dynamic Teacher (DDT) with Asynchronous EMA (AEMA) is proposed, which implements group-wise parameter updates. In contrast to traditional EMA, which simultaneously updates all parameters, AEMA dynamically decomposes teacher parameters into two functional groups based on their contributions to capture the domain shift. By applying a distinct smoothing coefficient to two groups, AEMA simultaneously enables fast adaptation and historical knowledge retention. Comprehensive experiments carried out on three widely used traffic benchmarks have demonstrated that the proposed DDT achieves superior performance, outperforming SOTA methods by a clear margin. The codes are available at https://github.com/qih96/DDT.