We report the successful formation of supported multicomponent lipid bilayer membranes (sLBMs) on polymer cushions consisting of a set of alternating maleic acid copolymers. The formation of sLBMs was triggered by a transient reduction of the electrostatic repulsion between the polymer cushions and the lipid vesicles by lowering the solution's pH to 4. Upon formation, the stability of the sLBMs was not affected by subsequent variations of the environmental pH. The degree of hydrophilicity and swelling of the anionic polymer cushions was found to determine both the kinetics of the membrane formation and the mobility of the lipid bilayer with lipid diffusion coefficients in the range from 0.26 to 2.6 Mm²S^-1. The introduced polymer cushion system is concluded to provide a versatile base for the integration of active transmembrane proteins in sLBMs.