In the context of Alzheimer's disease (AD), state-of-the-art methods separating normal control (NC) from AD patients or CN from progressive MCI (mild cognitive impairment patients converting to AD) achieve decent classification rates. However, they all perform poorly at separating stable MCI (MCI patients not converting to AD) and progressive MCI. Instead of using features extracted from a single temporal point, we address this problem using descriptors of the hippocampus evolutions between two time points. To encode the transformation, we use the framework of large deformations by diffeomorphisms that provides geodesic evolutions. To perform statistics on those local features in a common coordinate system, we introduce an extension of the K\"{a}rcher mean algorithm that defines the template modulo rigid registrations, and an initialization criterion that provides a final template leading to better matching with the patients. Finally, as local descriptors transported to this template do not directly perform as well as global descriptors (e.g. volume difference), we propose a novel strategy combining the use of initial momentum from geodesic shooting, extended K\"{a}rcher algorithm, density transport and integration on a hippocampus subregion, which is able to outperform global descriptors.