It is a bit hard to tell, but it seems that “MMP” atlas is one of the early attempts to move the HCP Glasser atlas into an MNI space. If one looks carefully, it has some defects like missing voxels in the cortical ribbon, lost clusters and voxels of regions away from the bulk of the atlas region. The version of the Glasser atlas we transformed to MNI 2009c seems a better alternative. I’ve put several posts on this forum and on the AFNI messageboard about our version, so that should be easy to find, and it’s distributed with AFNI.
Moving atlases between spaces, including native subject spaces, always has some degree of loss and gain of region size. For small regions, this will be more dramatic in percentage. Try to see the affected regions individually and in the context of the neighboring regions to see if the results look reasonable. Downsampling would make this issue even more obvious. For datasets with similar resolution, I often move atlases with the nearest neighbor interpolation approach and then follow with a kind of regularization I call modal smoothing to replace voxels with the most common label in a small neighborhood. That neighborhood is often about the cube root of 3 (~1.8 voxel dimensions) for a simple 27-voxel neighborhood of nearest neighbors that include face, edge and corner voxels.
ANTs has a couple alternative interpolation methods (from the ITK library) besides NearestNeighbor with the similar goal of regularizing output that might be useful for moving atlases - GenericLabel and MultiLabel.