  We study the limiting distribution of dense orbits of a lattice subgroup
$\Gamma\le \text{SL}(m+1,\mathbb{R})$ acting on
$H\backslash\text{SL}(m+1,\mathbb{R})$, with respect to a filtration of growing
norm balls. The novelty of our work is that the groups $H$ we consider have
infinitely many non-trivial connected components. For a specific such $H$, the
homogeneous space $H\backslash G$ identifies with $X_{m,m+1}$, a moduli space
of rank $m$-discrete subgroups in $\mathbb{R}^{m+1}$. This study is motivated
by the work of Shapira-Sargent who studied random walks on $X_{2,3}$.
