Numerous unsupervised domain adaptation (UDA) methods for bearing fault diagnosis rely on extracting domain-invariant features to mitigate the impact of domain shift interference. However, the lack of evaluation criteria results in limited interpretability of domain-invariant features. Additionally, current pseudo-label prediction methods heavily rely on label information or computational resources, and the traditional Softmax function fails to capture valuable information. To address these problems, this article proposes a UDA method based on domain-invariant features evaluation and knowledge distillation (KD) for bearing fault diagnosis. First, mutual information and soft attention mechanism are integrated into the extraction of multivariate features to access the quality of domain-invariant features and enhance interpretability. Then, the concept of KD is introduced to predict pseudo-labels in the target domain without relying on label information or computational resources. Furthermore, an asynchronous feature metric adaptive strategy is developed to adjust the feature alignment metric, considering the maturity and precision of pseudo-labels. The effectiveness and superiority of the proposed method are demonstrated through comparative experiments and ablation studies on two bearing datasets.