In modern manufacturing, process monitoring is crucial in ensuring the stability and safety of production processes. However, the frequent changes in industrial conditions necessitate timely updates and retraining of the on-site deployment of data-driven process monitoring methods, which is a task unattainable with the limited computational resources of edge devices. To address this issue, this paper proposes an interpretable switching deep Markov model (ISDMM)-based cloud-edge collaborative framework for industrial process monitoring. ISDMM defines discrete switching variables that represent working conditions and corresponding multiple transition networks. The transition networks are concurrently trained and switched based on the current working condition, enabling ISDMM to capture different dynamics of the system under different conditions. Before deployment, model simplification is performed to construct a model base where each model is tailored to individual working conditions. Moreover, in the edge layer, a model update strategy is designed to determine whether the model can be invoked from the model library based on the current working condition, alleviating the burden of unnecessary retraining. Finally, the effectiveness of ISDMM and its cloud-edge collaborative framework is validated through a numerical example and a real-world industrial process.