A conveyor device for conveying flexible containers along a packaging line, comprising: a track defining a closed path; at least one pair of movers (2a, 2b) including a left mover and a right mover movably mounted on the track for moving along the path; active and reactive (29, 30) elements associated with the track and the left and right movers (2a, 2b) and configured for independently driving and controlling the left and right movers (2a, 2b) along the path; at least one pair of gripper arms (3a, 3b) including a left gripper arm connected to the left mover and a right gripper arm connected to the right mover; at least one pair of clamps (4a, 4b) including a left clamp mounted on the left gripper arm and a right clamp mounted on the right gripper arm, the left and right clamps (4a, 4b) being configured for gripping a flexible container at opposite side edges thereof with the flexible container in a vertical position; wherein the closed path defined by the track lays in a horizontal plane and includes straight and curved sections; and the left gripper arm is pivotally connected to the left mover by a left articulation about a vertical axis and the right gripper arm is pivotally connected to the right mover by a right articulation about a vertical axis; the conveyor device being characterized in that: the left and right gripper arms (3a, 3b) are linked together by a parallelism-keeping mechanism configured for keeping the left and right gripper arms (3a, 3b) mutually parallel and with the left and right clamps (4a, 4b) facing each other in both the straight and curved sections of the path even when a distance between the left and right articulations (6a, 6b) is varied due to the relative movements of the left and right movers (2a, 2b). a track defining a closed path; at least one pair of movers (2a, 2b) including a left mover and a right mover movably mounted on the track for moving along the path; active and reactive (29, 30) elements associated with the track and the left and right movers (2a, 2b) and configured for independently driving and controlling the left and right movers (2a, 2b) along the path; at least one pair of gripper arms (3a, 3b) including a left gripper arm connected to the left mover and a right gripper arm connected to the right mover; at least one pair of clamps (4a, 4b) including a left clamp mounted on the left gripper arm and a right clamp mounted on the right gripper arm, the left and right clamps (4a, 4b) being configured for gripping a flexible container at opposite side edges thereof with the flexible container in a vertical position; wherein the closed path defined by the track lays in a horizontal plane and includes straight and curved sections; and the left gripper arm is pivotally connected to the left mover by a left articulation about a vertical axis and the right gripper arm is pivotally connected to the right mover by a right articulation about a vertical axis; the conveyor device being characterized in that: the left and right gripper arms (3a, 3b) are linked together by a parallelism-keeping mechanism configured for keeping the left and right gripper arms (3a, 3b) mutually parallel and with the left and right clamps (4a, 4b) facing each other in both the straight and curved sections of the path even when a distance between the left and right articulations (6a, 6b) is varied due to the relative movements of the left and right movers (2a, 2b).