An apparatus usable in an envelope stuffing machine for changing an initial advancement direction of piles made up of inserts to be stuffed in envelopes, comprising: at least a thrust device (31, 32, 33, 34) for pushing a pile in a thrust direction, arrangeable in such as way that the thrust direction is perpendicular or oblique with respect to the initial advancement direction, and at least a connecting device comprising drawing means (41, 42) for transporting piles along a transport pathway (P1, P2, P3), at a terminal end of which transport pathway there is an outlet of the connecting device, which drawing means (41, 42) are suitable for receiving piles which are supplied to the apparatus having said initial advancement direction, the drawing means (41, 42) being curved in such a way that the transport pathway (P1, P2, P3) comprises at least an arced tract with a centre of curvature located on a main axis perpendicular to the initial advancement direction, whereby, upon the crossing of the arced tract, the piles exhibit an intermediate advancement direction which is oblique or transversal with respect to the initial advancement direction; characterized in that the thrust device (31, 32, 33, 34) is located downstream of the connecting device in a position in which it can receive the piles which exit from the connecting device, the thrust device comprising at least two thrust elements (31, 32) rotatable about a respective vertical rotation axis, and each having a thrust surface (310, 320), the two thrust elements being arranged with respect to one another in such a way that, upon the activation in counter-rotation of the thrust elements (31, 32), the respective thrust surfaces (310, 320) are cyclically facing one another and at such a distance as to be able together to grab and push a pile in said thrust direction; wherein the thrust elements (31, 32) are located below the connecting device, and in that the thrust device comprises a fixed lower base located below the thrust surfaces (310, 320), and suitable for restingly receiving a side of the pile which falls between the two thrust elements. at least a thrust device (31, 32, 33, 34) for pushing a pile in a thrust direction, arrangeable in such as way that the thrust direction is perpendicular or oblique with respect to the initial advancement direction, and at least a connecting device comprising drawing means (41, 42) for transporting piles along a transport pathway (P1, P2, P3), at a terminal end of which transport pathway there is an outlet of the connecting device, which drawing means (41, 42) are suitable for receiving piles which are supplied to the apparatus having said initial advancement direction, the drawing means (41, 42) being curved in such a way that the transport pathway (P1, P2, P3) comprises at least an arced tract with a centre of curvature located on a main axis perpendicular to the initial advancement direction, whereby, upon the crossing of the arced tract, the piles exhibit an intermediate advancement direction which is oblique or transversal with respect to the initial advancement direction; characterized in that the thrust device (31, 32, 33, 34) is located downstream of the connecting device in a position in which it can receive the piles which exit from the connecting device, the thrust device comprising at least two thrust elements (31, 32) rotatable about a respective vertical rotation axis, and each having a thrust surface (310, 320), the two thrust elements being arranged with respect to one another in such a way that, upon the activation in counter-rotation of the thrust elements (31, 32), the respective thrust surfaces (310, 320) are cyclically facing one another and at such a distance as to be able together to grab and push a pile in said thrust direction; wherein the thrust elements (31, 32) are located below the connecting device, and in that the thrust device comprises a fixed lower base located below the thrust surfaces (310, 320), and suitable for restingly receiving a side of the pile which falls between the two thrust elements.