In a recent study based on homogeneous barium abundance measurements in open clusters (OCs), a trend of increasing Ba/Fe ratios for decreasing cluster age was reported. We present here further abundance determinations, relative to four other elements having important s-process contributions, with the aim of investigating whether or not the growth found for Ba/Fe is indicative of a general property, shared also by the other heavy elements formed by slow neutron captures. In particular, we derived abundances for yttrium, zirconium, lanthanum, and cerium, using equivalent width measurements and the MOOG code. Our sample includes 19 OCs of different ages, for which the spectra were obtained by the ESO Very Large Telescope using the UVES spectrometer. The growth previously suggested for Ba is confirmed for all the elements analyzed in our study. This fact implies significant changes in our views of the Galactic chemical evolution for elements beyond iron. Our results necessarily require that very low mass asymptotic giant branch stars (M1.5 M ) produce larger amounts of s-process elements (and hence activate the 13C-neutron source more effectively) than previously expected. Their role in producing neutron-rich elements in the Galactic disk has been so far underestimated, and their evolution and neutron-capture nucleosynthesis should now be reconsidered.