A plant for building tires for vehicle wheels, includes at least one forming drum having a supporting shaft axially projecting from opposite sides relative to a central body. A handling device is adapted to engage the forming drum and transfer it between at least one loading/unloading location and a work location provided for carrying out at least one process working on the forming drum. The handling device includes a grip assembly carrying at least two grip members each engaging the supporting shaft in the vicinity of each of the end portions thereof, and at least one motor associated with at least one of said grip members for driving the forming drum in rotation while the at least one process working is being carried out in the work location.