Privacy concerns make it infeasible to construct a large medical image dataset by fusing small ones from different sources/institutions. Therefore, federated learning (FL) becomes a promising technique to learn from multi-source decentralized data with privacy preservation. However, the cross-client variation problem in medical image data would be the bottleneck in practice. In this paper, we propose a variation-aware federated learning (VAFL) framework, where the variations among clients are minimized by transforming the images of all clients onto a common image space. We first select one client with the lowest data complexity to define the target image space and synthesize a collection of images through a privacy-preserving generative adversarial network, called PPWGAN-GP. Then, a subset of those synthesized images, which effectively capture the characteristics of the raw images and are sufficiently distinct from any raw image, is automatically selected for sharing with other clients. For each client, a modified CycleGAN is applied to translate its raw images to the target image space defined by the shared synthesized images. In this way, the cross-client variation problem is addressed with privacy preservation. We apply the framework for automated classification of clinically significant prostate cancer and evaluate it using multi-source decentralized apparent diffusion coefficient (ADC) image data. Experimental results demonstrate that the proposed VAFL framework stably outperforms the current horizontal FL framework. As VAFL is independent of deep learning architectures for classification, we believe that the proposed framework is widely applicable to other medical image classification tasks.