The machine (1) includes at least one assistance subassembly (10) having: an endless chain or belt (16); a plurality of thrust members (20) connected to the chain or belt (16) in succession and projecting from the chain or belt (16), and a drive device (21) of this chain or belt (16), several thrust members (20) are brought into the path (3) of the sliders (8) connected to the grippers (4) for gripping the film (F), so that at least one of the thrust members (20) comes to bear against at least one of these sliders (8) and exerts a thrust on this slider to assist the displacement of this slider along the rail (5e) on which this slider moves.