\v Cech Persistence diagrams (PDs) are topological descriptors routinely used to capture the geometry of complex datasets. They are commonly compared using the Wasserstein distances $\OT_p$; however, the extent to which PDs are stable with respect to these metrics remains poorly understood. We partially close this gap by focusing on the case where datasets are sampled on an $m$-dimensional submanifold of $\R^d$. Under this manifold hypothesis, we show that convergence with respect to the $\OT_p$ metric happens exactly when $p>m$. We also provide improvements upon the bottleneck stability theorem in this case and prove new laws of large numbers for the total $\alpha$-persistence of PDs. Finally, we show how these theoretical findings shed new light on the behavior of the feature maps on the space of PDs that are used in ML-oriented applications of Topological Data Analysis.