  In this work we study the long-time behavior for subcritical measure-valued
branching processes with immigration on the space of tempered measures. Under
some reasonable assumptions on the spatial motion, the branching and
immigration mechanisms, we prove the existence and uniqueness of an invariant
measure for the corresponding Markov transition semigroup. Moreover, we show
that it converges with exponential rate to the unique invariant measure in the
Wasserstein distance as well as in a distance defined in terms of Laplace
transforms. Finally, we consider an application of our results to super-L\'evy
processes as well as branching particle systems on the lattice with noncompact
spins.
