  Given a complete, connected Riemannian manifold $ \mathbb{M}^n $ with Ricci
curvature bounded from below, we discuss the stability of the solutions of a
porous medium-type equation with respect to the 2-Wasserstein distance. We
produce (sharp) stability estimates under negative curvature bounds, which to
some extent generalize well-known results by Sturm and Otto-Westdickenberg. The
strategy of the proof mainly relies on a quantitative $L^1-L^\infty$ smoothing
property of the equation considered, combined with the Hamiltonian approach
developed by Ambrosio, Mondino and Savar\'e in a metric-measure setting.
