  Let $X$ be a compact complex manifold in the Fujiki class $\mathscr{C}$. We
study the compactification of $\operatorname{Aut}^0(X)$ given by its closure in
Barlet cycle space. The boundary points give rise to non-dominant meromorphic
self-maps of $X$. Moreover convergence in cycle space yields convergence of the
corresponding meromorphic maps. There are analogous compactifications for
reductive subgroups acting trivially on $\operatorname{Alb} X$. If $X$ is
K\"ahler, these compactifications are projective. Finally we give applications
to the action of $\operatorname{Aut}(X)$ on the set of probability measures on
$X$. In particular we obtain an extension of Furstenberg lemma to manifolds in
the class $\mathscr{C}$.
