The parameter size of modern large language models (LLMs) can be scaled up via the sparsely-activated Mixture-of-Experts (MoE) technique to avoid excessive increase of the computational costs. To further improve training efficiency, pipelining computation and communication has become a promising solution for distributed MoE training. However, existing work primarily focuses on scheduling tasks within the MoE layer, such as expert computing and all-to-all (A2A) communication, while neglecting other key operations including multi-head attention (MHA) computing, gating, and all-reduce communication. In this paper, we propose FlowMoE, a scalable framework for scheduling multi-type task pipelines. First, FlowMoE constructs a unified pipeline to consistently scheduling MHA computing, gating, expert computing, and A2A communication. Second, FlowMoE introduces a tensor chunk-based priority scheduling mechanism to overlap the all-reduce communication with all computing tasks. We implement FlowMoE as an adaptive and generic framework atop PyTorch. Extensive experiments with 675 typical MoE layers and four real-world MoE models across two GPU clusters demonstrate that our proposed FlowMoE framework outperforms state-of-the-art MoE training frameworks, reducing training time by 13%-57%, energy consumption by 10%-39%, and memory usage by 7%-32%.