  We investigate via LDA+DMFT (local density approximation combined with
dynamical mean field theory) the manifestation of correlation effects in a wide
range of binding energies in the hole-doped family of Fe-pnictides
$A$Fe$_2$As$_2$ ($A={\rm K}$, Rb, Cs) as well as the fictitious FrFe$_2$As$_2$
and $a$-axis stretched CsFe$_2$As$_2$. This choice of systems allows for a
systematic analysis of the interplay of Hund's coupling $J_H$ and on-site
Coulomb repulsion $U$ in multi-orbital Fe-pnictides under negative pressure.
With increasing ionic size of the alkali metal, we observe a non-trivial change
in the iron $3d$ hoppings, an increase of orbitally-selective correlations and
the presence of incoherent weight at high-binding energies that do not show the
typical lower Hubbard-band behavior but rather characteristic features of a
Hund's metal. This is especially prominent in $a$-stretched CsFe$_2$As$_2$. We
also find that the coherent/incoherent electronic behavior of the systems is,
apart from temperature, strongly dependent on $J_H$ and we provide estimates of
the coherence scale $T^*$. We discuss these results in the framework of
reported experimental observations.
