We present a new multiscale algorithm for solving regular-ized Optimal Transport problems on the GPU, with a linear memory footprint. Relying on Sinkhorn divergences which are convex, smooth and positive definite loss functions, this method enables the computation of transport plans between millions of points in a matter of minutes. We show the effectiveness of this approach on brain tractograms modeled either as bundles of fibers or as track density maps. We use the resulting smooth assignments to perform label transfer for atlas-based segmentation of fiber tractograms. The parameters-blur and reach-of our method are meaningful, defining the minimum and maximum distance at which two fibers are compared with each other. They can be set according to anatomical knowledge. Furthermore, we also propose to estimate a probabilistic atlas of a population of track density maps as a Wasserstein barycenter. Our CUDA implementation is endowed with a user-friendly PyTorch interface, freely available on the PyPi repository (pip install geomloss) and at www.kernel-operations.io/geomloss.