A process for manufacturing tires for vehicle wheels includes building on a forming support a carcass structure including at least one carcass ply, and, at at least one end edge of the carcass ply, at least one annular anchoring structure including at least one substantially circumferential annular insert and at least one filling insert associated with the at least one substantially circumferential annular insert. Building the carcass structure includes positioning the annular anchoring structure on the carcass ply by moving the substantially circumferential annular insert into contact with an end edge of the carcass ply by means of a special positioning member, pulling down the filling insert on the end edge of the carcass ply and turning up a free end portion of the end edge of the carcass ply so as to form a turned up end portion of the carcass ply including the annular anchoring structure. The pulling down of the filling insert on the end edge of the carcass ply is carried out by a special pulling down member while the substantially circumferential annular insert is retained in a contact position with the end edge of the carcass ply by the positioning member.