A web transport assembly for moving flexible sheet material in a manufacturing process. The web transport assembly comprises first and second gripper chains spaced apart from one another to define a path of travel of the flexible sheet material, each of the gripper chains having a plurality of saddle links and a plurality of gripper links. First and second parallel rail structures are respectively aligned with the first and second gripper chains. In addition, upstream and downstream hubs are associated with each of the first and second gripper chains to define a continuous loop about which the associated gripper chain extends. At least one of the hubs includes a driven sprocket having a plurality of teeth engaging the associated gripper chain. The saddle links have a pair of spaced apart extension portions defining a spaced in which an associated one of the rails is received. The gripper links have a gripper mechanism that is configured to clamp a side of the flexible sheet material.