Fault diagnosis of mechanical equipment provides robust support for industrial production. It is worth noting that, the operation of mechanical equipment is accompanied by changes in factors such as speed and load, leading to significant differences in data distribution, which pose challenges for fault diagnosis. Additionally, in terms of application deployment, commonly used cloud-based fault diagnosis methods often encounter issues such as time delays and data security concerns, while common fault diagnosis methods cannot be directly applied to edge computing devices. Therefore, conducting fault diagnosis under cross-operating conditions based on edge computing holds significant research value. This paper proposes a domain-adaptation-based lightweight fault diagnosis framework tailored for edge computing scenarios. Incorporating the local maximum mean discrepancy into knowledge transfer aligns the feature distributions of different domains in a high-dimensional feature space, to discover a common feature space across domains. The acquired fault diagnosis expertise from the cloud-based deep neural network model is transferred to the lightweight edge-based model (edge model) using adaptation knowledge transfer methods. It aims to achieve accurate fault diagnosis under cross-working conditions while ensuring real-time diagnosis capabilities. We utilized the NVIDIA Jetson Xavier NX kit as the edge computing platform and conducted validation experiments on two devices. In terms of diagnostic performance, the proposed method significantly improved diagnostic accuracy, with average increases of 34.44% and 17.33% compared to existing methods, respectively.