  We show that the Morse boundary of a Morse local-to-global group is
$\sigma$-compact. Moreover, we show that the converse holds for small
cancellation groups. As an application, we show that the Morse boundary of a
non-hyperbolic, Morse local-to-global group that has contraction does not admit
a non-trivial stationary measure. In fact, we show that any stationary measure
on a geodesic boundary of such a groups needs to assign measure zero to the
Morse boundary. Unlike previous results, we do not need any assumptions on the
stationary measures considered.
