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The ability to combine data across scanners and studies is vital for neuroimaging, to increase both statistical power and the representation of biological variability. However, combining datasets across sites leads to two challenges: first, an increase in undesirable non-biological variance due to scanner and acquisition differences - the harmonisation problem - and second, data privacy concerns due to the inherently personal nature of medical imaging data, meaning that sharing them across sites may risk violation of privacy laws. To overcome these restrictions, we propose FedHarmony: a harmonisation framework operating in the federated learning paradigm. We show that to remove the scanner-specific effects, we only need to share the mean and standard deviation of the learned features, helping to protect individual subjects' privacy. We demonstrate our approach across a range of realistic data scenarios, using real multi-site data from the ABIDE dataset, thus showing the potential utility of our method for MRI harmonisation across studies. Our code is available at https://github.com/nkdinsdale/FedHarmony.