A press has a transfer direction along which workpieces are transferred through machining stations by gripper rails. Two carriages are provided per gripper rail which can be moved transversely to the transfer direction toward and away from one another. Two parallel control arms are rotatably disposed on one carriage which, on the other end, are rotatably disposed on a support part. On the other carriage, a link is rotatably disposed which, on the other end, is applied at half length to a parallel control arm. The support part can be lifted vertically by the movement of the carriages.