Joint extraction of entities and their relations benefits from the close interaction between named entities and their relation information. Therefore, how to effectively model such cross-modal interactions is critical for the final performance. Previous works have used simple methods, such as label-feature concatenation, to perform coarse-grained semantic fusion among cross-modal instances but fail to capture fine-grained correlations over token and label spaces, resulting in insufficient interactions. In this article, we propose a dynamic cross-modal attention network (CMAN) for joint entity and relation extraction. The network is carefully constructed by stacking multiple attention units in depth to dynamic model dense interactions over token-label spaces, in which two basic attention units and a novel two-phase prediction are proposed to explicitly capture fine-grained correlations across different modalities (e.g., token-to-token and label-to-token). Experiment results on the CoNLL04 dataset show that our model obtains state-of-the-art results by achieving 91.72% F1 on entity recognition and 73.46% F1 on relation classification. In the ADE and DREC datasets, our model surpasses existing approaches by more than 2.1% and 2.54% F1 on relation classification. Extensive analyses further confirm the effectiveness of our approach.