Convergence of locally homogeneous spaces

We study three different topologies on the moduli space $\mathscr{H}^{\rm loc}_m$ of equivariant isometry classes of $m$-dimensional locally homogeneous Riemannian spaces. As an application, we provide the first examples of locally homogeneous spaces converging to a limit space in the pointed $\mathcal{C}^{k,\alpha}$-topology, for some $k>1$, which do not admit any convergent subsequence in the pointed $\mathcal{C}^{k+1}$-topology.