A damping device for a leaf of a lift-and-slide window or door, said damping device being configured to damp the movement of the leaf in the opening and/or closing direction and comprising: - a containment body solidly connectable to the leaf; - a cursor slidably inserted into the containment body and movable reversibly between at least a first and a second position along a direction substantially parallel to a movement direction of the leaf; characterized in that the cursor comprises a first and a second abutting element (4, 5) adapted to engage with a striker element present on the fixed frame, said abutting elements (4, 5) facing one another so that the first abutting element engages the striker element during a passage of the cursor from the second position to the first position and the second abutting element engages the striker element during a passage of the cursor from the first position to the second position; wherein each of said abutting element is further reversibly switchable between a raised condition, in which it protrudes at least partially from said cursor to abut on the striker element, and a lowered condition, in which it is at least partially housed inside the cursor to permit a relative motion between the cursor and the striker element; - a cam profile arranged on the containment body and operationally active on the abutting elements to activate switching thereof from the raised condition to the lowered condition and vice versa following movement of the cursor between the first and the second position along the containment body; - damping means interposed between the containment body and the cursor. - a containment body solidly connectable to the leaf; - a cursor slidably inserted into the containment body and movable reversibly between at least a first and a second position along a direction substantially parallel to a movement direction of the leaf; characterized in that the cursor comprises a first and a second abutting element (4, 5) adapted to engage with a striker element present on the fixed frame, said abutting elements (4, 5) facing one another so that the first abutting element engages the striker element during a passage of the cursor from the second position to the first position and the second abutting element engages the striker element during a passage of the cursor from the first position to the second position; wherein each of said abutting element is further reversibly switchable between a raised condition, in which it protrudes at least partially from said cursor to abut on the striker element, and a lowered condition, in which it is at least partially housed inside the cursor to permit a relative motion between the cursor and the striker element; characterized in that the cursor comprises a first and a second abutting element (4, 5) adapted to engage with a striker element present on the fixed frame, said abutting elements (4, 5) facing one another so that the first abutting element engages the striker element during a passage of the cursor from the second position to the first position and the second abutting element engages the striker element during a passage of the cursor from the first position to the second position; wherein each of said abutting element is further reversibly switchable between a raised condition, in which it protrudes at least partially from said cursor to abut on the striker element, and a lowered condition, in which it is at least partially housed inside the cursor to permit a relative motion between the cursor and the striker element; - a cam profile arranged on the containment body and operationally active on the abutting elements to activate switching thereof from the raised condition to the lowered condition and vice versa following movement of the cursor between the first and the second position along the containment body; - damping means interposed between the containment body and the cursor.