In this paper, we introduce Modality-Inconsistent Continual Learning (MICL), a new continual learning scenario for Multimodal Large Language Models (MLLMs) that involves tasks with inconsistent modalities (image, audio, or video) and varying task types (captioning or question-answering). Unlike existing vision-only or modality-incremental settings, MICL combines modality and task type shifts, both of which drive catastrophic forgetting. To address these challenges, we propose MoInCL, which employs a Pseudo Targets Generation Module to mitigate forgetting caused by task type shifts in previously seen modalities. It also incorporates Instruction-based Knowledge Distillation to preserve the model's ability to handle previously learned modalities when new ones are introduced. We benchmark MICL using a total of six tasks and conduct experiments to validate the effectiveness of our MoInCL. The experimental results highlight the superiority of MoInCL, showing significant improvements over representative and state-of-the-art continual learning baselines.