Fact verification is an essential tool to mitigate the spread of false information online, which has gained a widespread attention recently. However, a fact verification in the question-answering dialogue is still underexplored. In this paper, we propose a neural network based approach called question-answering dialogue based fact verification with mixture of experts (QaDialMoE). It exploits questions and evidence effectively in the verification process and can significantly improve the performance of fact verification. Specifically, we exploit the mixture of experts to focus on various interactions among responses, questions and evidence. A manager with an attention guidance module is implemented to guide the training of experts and assign a reasonable attention score to each expert. A prompt module is developed to generate synthetic questions that make our approach more generalizable. Finally, we evaluate the QaDialMoE and conduct a comparative study on three benchmark datasets. The experimental results demonstrate that our QaDialMoE outperforms previous approaches by a large margin and achieves new state-of-the-art results on all benchmarks. This includes the accuracy improvements on the HEALTHVER as 84.26%, the FAVIQ A dev set as 78.7%, the FAVIQ R dev set as 86.1%, test set as 86.0%, and the COLLOQUIAL as 89.5%. To our best knowledge, this is the first work to investigate a question-answering dialogue based fact verification, and achieves new state-of-the-art results on various benchmark datasets.