One-sample and multi-sample tests on the concentration parameter of Fisher-von Mises-Langevin distributions on (hyper-)spheres have been well studied in the literature. However, only little is known about their behaviour under local alternatives, which is due to complications inherent to the curved nature of the parameter space. The aim of the present paper therefore consists in filling that gap by having recourse to the Le Cam methodology, which has recently been adapted from the linear to the spherical setup. We obtain explicit expressions of the powers for the most efficient one- and multi-sample tests. As a nice by-product, we are also able to write down the powers (against local Fisher-von Mises-Langevin alternatives) of the celebrated Rayleigh test of uniformity. A Monte Carlo simulation study confirms our theoretical findings and shows the empirical powers of the above-mentioned procedures.