Motor imagery brain-computer interface (MI-BCI) can parse user motor imagery to achieve wheelchair control or motion control for smart prostheses. However, problems of poor feature extraction and low cross-subject performance exist in the model for motor imagery classification tasks. To address these problems, we propose a multi-scale adaptive transformer network (MSATNet) for motor imagery classification. Therein, we design a multi-scale feature extraction (MSFE) module to extract multi-band highly-discriminative features. Through the adaptive temporal transformer (ATT) module, the temporal decoder and multi-head attention unit are used to adaptively extract temporal dependencies. Efficient transfer learning is achieved by fine-tuning target subject data through the subject adapter (SA) module. Within-subject and cross-subject experiments are performed to evaluate the classification performance of the model on the BCI Competition IV 2a and 2b datasets. The MSATNet outperforms benchmark models in classification performance, reaching 81.75 and 89.34% accuracies for the within-subject experiments and 81.33 and 86.23% accuracies for the cross-subject experiments. The experimental results demonstrate that the proposed method can help build a more accurate MI-BCI system.