Dynamic multimodal multiobjective optimization presents the dual challenge of simultaneously tracking multiple equivalent pareto optimal sets and maintaining population diversity in time-varying environments. However, existing dynamic multiobjective evolutionary algorithms often neglect solution modality, whereas static multimodal multiobjective evolutionary algorithms lack adaptability to dynamic changes. To address above challenge, this paper makes two primary contributions. First, we introduce a new benchmark suite of dynamic multimodal multiobjective test functions constructed by fusing the properties of both dynamic and multimodal optimization to establish a rigorous evaluation platform. Second, we propose a novel algorithm centered on a Clustering-based Autoencoder prediction dynamic response mechanism, which utilizes an autoencoder model to process matched clusters to generate a highly diverse initial population. Furthermore, to balance the algorithm's convergence and diversity, we integrate an adaptive niching strategy into the static optimizer. Empirical analysis on 12 instances of dynamic multimodal multiobjective test functions reveals that, compared with several state-of-the-art dynamic multiobjective evolutionary algorithms and multimodal multiobjective evolutionary algorithms, our algorithm not only preserves population diversity more effectively in the decision space but also achieves superior convergence in the objective space.