In a process for building tires for vehicle wheels, a carcass sleeve is arranged on a shaping drum including two half-drums and two supports. Each support is operatively associated with one of the half-drums and carries a plurality of turning-up levers having free ends positioned close to the respective half-drum. Two annular anchoring structures are disposed around the carcass sleeve and each of the two supports is locked to the respective half-drum by interposing only stiff mechanical elements in mutual contact between the support and the half-drum. The carcass sleeve is shaped in a toroidal conformation, by setting a threaded shaft in rotation and moving the half-drums dose to each other. Subsequently, each of the two supports is unlocked from the respective half-drum and the threaded shaft is set in rotation to move each support close to the respective half-drum and lift up the free ends of the turning-up levers until the end flaps of the carcass sleeve are turned up around the annular anchoring structures.