  By performing pressure simulations within density functional theory for the
family of iron-based superconductors $Ae{}A$Fe$_4$As$_4$ with $Ae$ = Ca, Sr, Ba
and $A$ = K, Rb, Cs we predict in these systems the appearance of two
consecutive half-collapsed tetragonal transitions at pressures $P_{c_1}$ and
$P_{c_2}$, which have a different character in terms of their effect on the
electronic structure. We find that, similarly to previous studies for
CaKFe$_4$As$_4$, spin-vortex magnetic fluctuations on the Fe sublattice play a
key role for an accurate structure prediction in these materials at zero
pressure. We identify clear trends of critical pressures and discuss the
relevance of the collapsed phases in connection to magnetism and
superconductivity. Finally, the intriguing cases of EuRbFe$_4$As$_4$ and
EuCsFe$_4$As$_4$, where Eu magnetism coexists with superconductivity, are
discussed as well in the context of half-collapsed phases.
