In real-world scenarios, multimodal information continuously evolves, with new entity and relation types emerging, necessitating timely updates to multimodal knowledge graphs for supporting downstream tasks. However, existing methods struggle to balance real-time adaptability and computational efficiency in continual learning scenarios. To this end, this paper proposes the Continual Multimodal Entity and Relation Joint Extraction (CMERJE) task and a Multimodal Prompt-based Boundary-enhanced Continual (MPBoCo) framework. Specifically, MPBoCo incrementally stores task-specific knowledge via learnable multimodal prompts, dynamically matches relevant prompts for each instance, and fuses them into a frozen backbone model for task-specific reasoning. Subsequently, the boundary-enhanced dual-branch module leverages the auxiliary branch to preserve local syntactic continuity and provide boundary guidance. Experimental results demonstrate that MPBoCo achieves superior performance in real-world scenarios, significantly outperforming baseline methods by 5.5% and 7.2% in 10-task and 5-task settings, respectively.