A device for supporting and transferring, in a shaping station (2), at least the central part of glass sheets which have been heated to their shaping temperature. The device includes a pressurized box (21) generating an air cushion. The box has a plurality of individual compartments (53), transverse with respect to the axis of travel of the glass sheets (28), each of the compartments (53) having a width measured parallel to said axis which is small by comparison with the "height" of the glass sheets and, being provided with gas having at least two pressure distribution levels.