An apparatus for packaging a product arranged on a support, said support having a base wall and a side wall, said apparatus comprising: a packaging assembly configured for tightly fixing one or more film sheets to one or more of the supports, the packaging assembly including: - a lower tool comprising at least an inner wall defining a prefixed number of seats for receiving the one or more supports, and - an upper tool facing the lower tool and comprising a film holding plate configured for holding the one or more film sheets, the film holding plate having a respective active surface configured for receiving the one or more film sheets, at least the upper and lower tools (21; 22) cooperating to define a packaging chamber; said packaging assembly being configured to operate at least in a first operating condition, where said packaging chamber is open to receive the one or more film sheets, and in a second operating condition, where said packaging chamber is closed, optionally hermetically closed; a film supplying assembly configured for supplying a continuous film; a film cutting assembly active on the continuous film and configured for cutting film sheets of prefixed length from said continuous film; and a control unit connected to the packaging assembly and configured for commanding the packaging assembly to pass from the first to the second operating condition and vice-versa, wherein the film holding plate has a lateral surface extending substantially transverse, optionally substantially perpendicular, to a plane defined by the active surface, characterized in that the film holding plate comprises a plurality of ejectors arranged in the lateral surface, each of the plurality of ejectors being configured for ejecting a stream of gas (31 b) in a direction substantially parallel to said plane and substantially away from a center of the active surface. a packaging assembly configured for tightly fixing one or more film sheets to one or more of the supports, the packaging assembly including: - a lower tool comprising at least an inner wall defining a prefixed number of seats for receiving the one or more supports, and - an upper tool facing the lower tool and comprising a film holding plate configured for holding the one or more film sheets, the film holding plate having a respective active surface configured for receiving the one or more film sheets, at least the upper and lower tools (21; 22) cooperating to define a packaging chamber; - a lower tool comprising at least an inner wall defining a prefixed number of seats for receiving the one or more supports, and - an upper tool facing the lower tool and comprising a film holding plate configured for holding the one or more film sheets, the film holding plate having a respective active surface configured for receiving the one or more film sheets, at least the upper and lower tools (21; 22) cooperating to define a packaging chamber; said packaging assembly being configured to operate at least in a first operating condition, where said packaging chamber is open to receive the one or more film sheets, and in a second operating condition, where said packaging chamber is closed, optionally hermetically closed; a film supplying assembly configured for supplying a continuous film; a film cutting assembly active on the continuous film and configured for cutting film sheets of prefixed length from said continuous film; and a control unit connected to the packaging assembly and configured for commanding the packaging assembly to pass from the first to the second operating condition and vice-versa, wherein the film holding plate has a lateral surface extending substantially transverse, optionally substantially perpendicular, to a plane defined by the active surface, characterized in that the film holding plate comprises a plurality of ejectors arranged in the lateral surface, each of the plurality of ejectors being configured for ejecting a stream of gas (31 b) in a direction substantially parallel to said plane and substantially away from a center of the active surface.