Federated learning has shown its unique advantages in many different tasks, including brain image analysis. It provides a new way to train deep learning models while protecting the privacy of medical image data from multiple sites. However, previous studies suggest that domain shift across different sites may influence the performance of federated models. As a solution, we propose a gradient matching federated domain adaptation (GM-FedDA) method for brain image classification, aiming to reduce domain discrepancy with the assistance of a public image dataset and train robust local federated models for target sites. It mainly includes two stages: 1) pretraining stage; we propose a one-common-source adversarial domain adaptation (OCS-ADA) strategy, i.e., adopting ADA with gradient matching loss to pretrain encoders for reducing domain shift at each target site (private data) with the assistance of a common source domain (public data) and 2) fine-tuning stage; we develop a gradient matching federated (GM-Fed) fine-tuning method for updating local federated models pretrained with the OCS-ADA strategy, i.e., pushing the optimization direction of a local federated model toward its specific local minimum by minimizing gradient matching loss between sites. Using fully connected networks as local models, we validate our method with the diagnostic classification tasks of schizophrenia and major depressive disorder based on multisite resting-state functional MRI (fMRI), respectively. Results show that the proposed GM-FedDA method outperforms other commonly used methods, suggesting the potential of our method in brain imaging analysis and other fields, which need to utilize multisite data while preserving data privacy.