Pnictides of the family AFe2As2, where A is an alkaline earth element, exhibit several phase transitions in their structure and magnetic order as functions of applied pressure. We employ density functional theory total energy calculations at T=0K to model these transitions for the entire set of alkaline earths (A=Ca, Sr, Ba, Ra) which form the 122 family. Three distinct types of transition occur: an enthalpic transition in which the striped antiferromagnetic orthorhombic (OR-AFM) phase swaps thermodynamic stability with a competing tetragonal phase; a soft-mode transition through which the OR-AFM phase loses its magnetism and orthorhombicity; a lattice parameter anomaly in which the tetragonal c-axis collapses. We identify this last transition as a ``Lifshitz transition'' caused by a change in Fermi surface topology. Depending on the element A, the tetragonal state exhibiting the Lifshitz transition might be metastable (A=Ca) or stable (A=Sr, Ba and Ra).