A process and an apparatus for building tyres, wherein, in the process, a carcass sleeve including at least one carcass ply and a pair of annular anchoring structures is transferred to a shaping drum including flange elements that are each engageable with one of said annular anchoring structures. The carcass sleeve is shaped according to a toroidal configuration. Before shaping, annular grip elements respectively carried by the flange elements are radially expanded, each inside one of the annular anchoring structures, in order to engage the carcass sleeve. The annular anchoring structures are moved axially apart from each other before engaging the carcass sleeve by the annular grip elements.