A system for advancement of a consumable in the form of a continuous sheet in a machine for manufacturing containers having a first pulling group of the sheet actuated by a first motor at the end of the manufacturing zone, a second pulling group of the sheet actuated by a second motor at the end of the manufacturing zone, a support and deviation element of the sheet after the second pulling group equipped with a load cell for estimating the tension, and a processor programmed to direct the individual actuation of the first and second motors, such that the pulling groups promote the advancement of longitudinal portions of sheet sufficiently different to correct deviations in the tension of the sheet with respect to a pre-established target tension. A method for intermittent advancement of the sheet having a correction phase of the tension wherein only the second motor is actuated.