Nowadays, misinformation articles, especially multimodal ones, are widely spread on social media platforms and cause serious negative effects. To control their propagation, Multimodal Misinformation Detection (MMD) becomes an active topic in the community to automatically identify misinformation. Previous MMD methods focus on supervising detectors by collecting offline data. However, in real-world scenarios, new events always continually emerge, making MMD models trained on offline data consistently outdated and ineffective. To address this issue, training MMD models under online data streams is an alternative, inducing an emerging task named continual MMD. Unfortunately, it is hindered by two major challenges. First, training on new data consistently decreases the detection performance on past data, named past knowledge forgetting. Second, the social environment constantly evolves over time, affecting the generalization on future data. To alleviate these challenges, we propose to remember past knowledge by isolating interference between event-specific parameters with a Dirichlet process-based mixture-of-expert structure, and anticipate future environmental distributions by learning a continuous-time dynamics model. Accordingly, we induce a new continual MMD method DAEDCMD. Extensive experiments demonstrate that DAEDCMD can consistently and significantly outperform the compared methods, including six MMD baselines and three continual learning methods.