  This note is concerned with two families of operators related to the
fractional Laplacian, the first arising from the Caffarelli-Silvestre extension
problem and the second from the fractional heat equation. They both include the
Poisson semigroup. We show that on a complete, connected, and non-compact
Riemannian manifold of non-negative Ricci curvature, in both cases, the
solution with $L^1$ initial data behaves asymptotically as the mass times the
fundamental solution. Similar long-time convergence results remain valid on
more general manifolds satisfying the Li-Yau two-sided estimate of the heat
kernel. The situation changes drastically on hyperbolic space, and more
generally on rank one non-compact symmetric spaces: we show that for the
Poisson semigroup, the convergence to the Poisson kernel fails -but remains
true under the additional assumption of radial initial data.
