  Let $A\subseteq B$ be a $C^*$-inclusion. We give efficient conditions under
which $A$ separates ideals in $B$, and $B$ is purely infinite if every positive
element in $A$ is properly infinite in $B$. We specialise to the case when $B$
is a crossed product for an inverse semigroup action by Hilbert bimodules or a
section $C^*$-algebra of a Fell bundle over an \'etale, possibly non-Hausdorff,
groupoid. Then our theory works provided $B$ is the recently introduced
essential crossed product and the action is essentially exact and residually
aperiodic or residually topologically free. These last notions are developed in
the article.
