We study the role of radial motions of stars and gas on the evolution of abundance profiles in the Milky Way disk. We investigate, in a parametrized way, the impact of radial flows of gas and radial migration of stars induced mainly by the Galactic bar and its iteraction with the spiral arms. We use a model with several new or up-dated ingredients (atomic and molecular gas phases, star formation depending on molecular gas, recent sets of metallicity-dependent stellar yields from H to Ni, observationally inferred SNIa rates), which reproduces well most global and local observables of the Milky Way. We obtain abundance profiles flattening both in the inner disk (because of radial flows) and in the outer disk (because of the adopted star formation law). The gas abundance profiles flatten with time, but the corresponding stellar profiles appear to be steeper for younger stars, because of radial migration. We find a correlation between the stellar abundance profiles and O/Fe, which is a proxy for stellar age. Our final abundance profiles are in overall agreement with observations, but slightly steeper (by 0.01-0.02 dex/kpc) for elements above S. We find an interesting odd-even effect in the behaviour of the abundance profiles (steeper slopes for odd elements) for all sets of stellar yields; however, this behaviour does not appear in observations, suggesting that the effect is, perhaps, overestimated in current stellar nucleosynthesis calculations.