At ambient conditions, the quaternary sulfides LiCuFeS2 and KCuFeS2 present totally different crystal structures: while LiCuFeS2 crystallizes in a trigonal CaAl2Si2-type structure, a tetragonal ThCr2Si2-like structure is found for KCuFeS2. In this work, we present a computational study describing first the changes in the structural preference of the ACuFe(2) phases as a function of the alkali ion and second, the structural stability of the CuFeS2 phases obtained by electrochemical removal of the alkali cations from the two ACuFeS(2) compounds. A high copper mobility is found to be responsible for the observed metastability of the layered trigonal CuFeS2 phase obtained by delithiation of LiCuFeS2. In contrast, the tetragonal CuFeS2 structure obtained removing potassium from KCuFeS2 is predicted to be stable, both from the kinetic and thermodynamic points of view. The possibility of stabilizing mixed LixCu1-xFeS2 phases with a ThCr2Si2-type structure and the mobility of lithium in these is also explored.