With the adoption of powerful machine learning methods in medical image analysis, it is becoming increasingly desirable to aggregate data that is acquired across multiple sites. However, the underlying assumption of many analysis techniques that corresponding tissues have consistent intensities in all images is often violated in multi-centre databases. We introduce a novel intensity normalisation scheme based on density matching, wherein the histograms are modelled as Dirichlet process Gaussian mixtures. The source mixture model is transformed to minimise its L2 divergence towards a target model, then the voxel intensities are transported through a mass-conserving flow to maintain agreement with the moving density. In a multi-centre study with brain MRI data, we show that the proposed technique produces excellent correspondence between the matched densities and histograms. We further demonstrate that our method makes tissue intensity statistics substantially more compatible between images than a baseline affine transformation and is comparable to state-of-the-art while providing considerably smoother transformations. Finally, we validate that nonlinear intensity normalisation is a step toward effective imaging data harmonisation.