  In this article we extend the notion of $L^p$-measured subgroups couplings; a
quantitative asymmetric version of measure equivalence that was introduced by
Delabie, Koivisto, Le Ma\^itre and Tessera for finitely generated groups, to
the setting of locally compact compactly generated unimodular groups. As an
example of these couplings; using ideas from Bader and Rosendal, we prove a
"dynamical criteria" for the existence of regular embeddings between amenable
locally compact compactly generated unimodular groups, namely the existence of
an $L^\infty$-measured subgroup coupling that is coarsely $m$ to $1$. We then
proceed to prove that the existence of an $L^p$-measured subgroup that is
coarsely $m$ to $1$ implies the monotonicity of the $L^p$-isoperimetric
profile. We conclude then that the $L^p$-isoperimetric profile is monotonous
under regular embeddings, as well as coarse embeddings, between amenable
unimodular locally compact compactly generated groups.
