Process for building tyres for vehicle wheels, comprising: arranging at least one semi-finished product comprising cords parallel to each other; cutting said semi-finished product along a cutting line parallel to the extension of the cords in order to obtain a strip section; circumferentially winding the strip section on a forming drum; wherein the action of cutting comprises: positioning said semi-finished product below at least one blade; supporting the semi-finished product by a grip action applied on an upper surface of the semi-finished product; translating the at least one blade along the cutting line, through the semi-finished product supported by said grip action, wherein the semi-finished product is positioned below the at least one blade with a lower surface of the semi-finished product in abutment relation on at least one slide plate arranged in an operative position; moving said slide plate away from the semi-finished product, after the activation of the grip action and before the actuation of said at least one blade along the cutting line, by translating the slide plate from an operative position in which it is coplanar with the advancement plane to a rest position in which it is arranged below the advancement plane. arranging at least one semi-finished product comprising cords parallel to each other; cutting said semi-finished product along a cutting line parallel to the extension of the cords in order to obtain a strip section; circumferentially winding the strip section on a forming drum; wherein the action of cutting comprises: positioning said semi-finished product below at least one blade; supporting the semi-finished product by a grip action applied on an upper surface of the semi-finished product; translating the at least one blade along the cutting line, through the semi-finished product supported by said grip action, wherein the semi-finished product is positioned below the at least one blade with a lower surface of the semi-finished product in abutment relation on at least one slide plate arranged in an operative position; moving said slide plate away from the semi-finished product, after the activation of the grip action and before the actuation of said at least one blade along the cutting line, by translating the slide plate from an operative position in which it is coplanar with the advancement plane to a rest position in which it is arranged below the advancement plane. positioning said semi-finished product below at least one blade; supporting the semi-finished product by a grip action applied on an upper surface of the semi-finished product; translating the at least one blade along the cutting line, through the semi-finished product supported by said grip action, wherein the semi-finished product is positioned below the at least one blade with a lower surface of the semi-finished product in abutment relation on at least one slide plate arranged in an operative position; moving said slide plate away from the semi-finished product, after the activation of the grip action and before the actuation of said at least one blade along the cutting line, by translating the slide plate from an operative position in which it is coplanar with the advancement plane to a rest position in which it is arranged below the advancement plane.