Method to move containers (C) in a processing line (10) that comprises a station (11) for storing and picking up said containers (C) and a station (12) at least for filling said containers (C), said method providing, in said storage and pick-up station (11), to supply a container-holding tray (15) containing a plurality of said containers (C) disposed upended in said container-holding tray (15) with respect to their mouth (43), and subsequently to upend said containers (C) with the aid of upending means (20) so as to invert the orientation of said containers (C) and make them available with the side of the mouth (43) facing upward so that they can be picked up by extraction means (22).