Automatic speech recognition (ASR) is the major human-machine interface in many intelligent systems, such as intelligent homes, autonomous driving, and servant robots. However, its performance usually significantly deteriorates in the presence of external noise, leading to limitations of its application scenes. The audio-visual speech recognition (AVSR) takes visual information as a complementary modality to enhance the performance of audio speech recognition effectively, particularly in noisy conditions. Recently, the transformer-based architectures have been used to model the audio and video sequences for the AVSR, which achieves a superior performance. However, its performance may be degraded in these architectures due to extracting irrelevant information while modeling long-term dependences. In addition, the motion feature is essential for capturing the spatio-temporal information within the lip region to best utilize visual sequences but has not been considered in the AVSR tasks. Therefore, we propose a multimodal sparse transformer network (MMST) in this article. The sparse self-attention mechanism can improve the concentration of attention on global information by selecting the most relevant parts wisely. Moreover, the motion features are seamlessly introduced into the MMST model. We subtly allow motion-modality information to flow into visual modality through the cross-modal attention module to enhance visual features, thereby further improving recognition performance. Extensive experiments conducted on different datasets validate that our proposed method outperforms several state-of-the-art methods in terms of the word error rate (WER).