  We consider classes of arbitrary (finite or infinite) graphs of bounded
shrub-depth, specifically the classes $\mathrm{TM}_r(d)$ of arbitrary graphs
that have tree models of height $d$ and $r$ labels. We show that the graphs of
$\mathrm{TM}_r(d)$ are $\mathrm{MSO}$-pseudo-finite relative to the class
$\mathrm{TM}^{\text{f}}_r(d)$ of finite graphs of $\mathrm{TM}_r(d)$; that is,
that every $\mathrm{MSO}$ sentence true in a graph of $\mathrm{TM}_r(d)$ is
also true in a graph of $\mathrm{TM}^{\text{f}}_r(d)$. We also show that
$\mathrm{TM}_r(d)$ is closed under ultraproducts and ultraroots. These results
have two consequences. The first is that the index of the
$\mathrm{MSO}[m]$-equivalence relation on graphs of $\mathrm{TM}_r(d)$ is
bounded by a $(d+1)$-fold exponential in $m$. The second is that
$\mathrm{TM}_r(d)$ is exactly the class of all graphs that are
$\mathrm{MSO}$-pseudo-finite relative to $\mathrm{TM}^{\text{f}}_r(d)$.
