We measure experimentally the rearrangements due to a small localized cyclic displacement applied to a packing of rigid grains under gravity in a 2D geometry. We analyze the evolution of the response to this perturbation by considering the individual particle displacement and the coarse grained displacement field, as well as the mean packing fraction and coordination number. We find that the displacement response is rather long ranged, and evolves considerably with the number of cycles. We show that a small difference in the preparation method (induced by tapping the container) leads to a significant modification in the response though the packing fraction changes are minute. Not only the initial response but also its further evolution change with preparation, demonstrating that the system still retains a memory of the initial preparation after many cycles. Nevertheless, after a sufficient number of cycles, the displacement response for both preparation methods converges to a nearly radial field with a 1/r decay from the perturbation source. The observed differences between the preparation methods seem to be related to the changes in the coordination number (which is more sensitive to the evolution of the packing than the packing fraction). Specifically, it may be understood as an effect of the breaking of local arches, which affects the lateral transmission of forces.