We show that we can obtain a good fit to the present-day stellar-mass functions of a large sample of young and old Galactic clusters with a tapered Salpeter power-law distribution function with an exponential truncation of the form dN/dm ∝ mα [1 - exp(-m/mc)β]. The average value of the power-law index α is ~-2.2, very close to the Salpeter value of -2.3, while the characteristic mass, mc, is in the range 0.1-0.6M⊙ and does not seem to vary in any systematic way with the present cluster parameters such as metal abundance, total cluster mass or central concentration. However, the characteristic mass shows a remarkable correlation with the dynamical age of the cluster, namely mc/M⊙ ≃ 0.15 + 0.5 × t3/4dyn, where tdyn is the dynamical time, taken as the ratio of cluster age and dissolution time. The small scatter around this correlation is likely due to uncertainties on the estimated value of tdyn. We attribute the observed trend to the onset of mass segregation through two-body relaxation in a tidal environment, causing preferential loss of low-mass stars from the cluster and hence a drift of the characteristic mass towards higher values. If dynamical evolution is indeed at the origin of the observed trend, it seems plausible that globular clusters, now with mc ≃ 0.35M⊙, were born with a stellar mass function very similar to that measured today in the youngest Galactic clusters and with a value of mc around 0.15 M⊙. This is consistent with the absence of a turn-over in the mass function of the Galactic bulge down to the observational limit at ~0.2M⊙ and argues for the universality of the initial mass function of Population I and II stars.