By performing pressure simulations within density functional theory for the family of iron-based superconductors AeAFe4As4 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 Pc1 and Pc2, which have a different character in terms of their effect on the electronic structure. We find that, similar to previous studies for CaKFe4As4, 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 EuRbFe4As4 and EuCsFe4As4, where Eu magnetism coexists with superconductivity, are discussed as well in the context of half-collapsed phases.