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Most domain adaptation methods for machine reading comprehension (MRC) use a pre-trained question-answer (QA) construction model to generate pseudo QA pairs for MRC transfer. Such a process will inevitably introduce mismatched pairs (i.e., noisy correspondence) due to i) the unavailable QA pairs in target documents, and ii) the domain shift during applying the QA construction model to the target domain. Undoubtedly, the noisy correspondence will degenerate the performance of MRC, which however is neglected by existing works. To solve such an untouched problem, we propose to construct QA pairs by additionally using the dialogue related to the documents, as well as a new domain adaptation method for MRC. Specifically, we propose Robust Domain Adaptation for Machine Reading Comprehension (RMRC) method which consists of an answer extractor (AE), a question selector (QS), and an MRC model. Specifically, RMRC filters out the irrelevant answers by estimating the correlation to the document via the AE, and extracts the questions by fusing the candidate questions in multiple rounds of dialogue chats via the QS. With the extracted QA pairs, MRC is fine-tuned and provides the feedback to optimize the QS through a novel reinforced self-training method. Thanks to the optimization of the QS, our method will greatly alleviate the noisy correspondence problem caused by the domain shift. To the best of our knowledge, this could be the first study to reveal the influence of noisy correspondence in domain adaptation MRC models and show a feasible way to achieve robustness to mismatched pairs. Extensive experiments on three datasets demonstrate the effectiveness of our method.