We investigate the flux of main-belt asteroid fragments into resonant orbits converting them into near-Earth asteroids (NEAs), and the variability of this flux due to chance interasteroidal collisions. A numerical model is used, based on collisional physics consistent with the results of laboratory impact experiments. The assumed main-belt asteroid size distribution is derived from that of known asteroids extrapolated down to sizes of S 40 cm, modified in such a way to yield a quasi-stationary fragment production rate over times ZZ 100 Myr. The results show that the asteroid belt can supply a few hundred km-sizedNEAs per year, well enough to sustain the current population of such bodies. On the other hand, if our collisional physics is correct, the number of existing IO-km objects implies that these objects either have very long-lived orbits, or must come from a different source (i.e., comet,s). Our model predicts that the fragments supplied from the asteroid belt have initially a power-law size distribution somewhat steeper than the observed one, suggesting preferential removal of small objects. The component of t$he NEA population with dynamical lifetimes shorter than or of the order of 1 Myr can vary by a factor reaching up to a few tens, due to single large-scale collisions in the main belt; these fluctuations are enhanced for smaller bodies and faster evolutionary time scales. As a consequence, the Earth's cratering rate can also change by about an order of ma.gnitude over the 0.1 to 1 Myr time scales. Despite these sporadic spikes, when averaged over times of 10 Myr or longer the fluctuat,ions are unlikely to exceed a factor two.