Abstract The dynamical lifetimes of small bodies against ejection from the Solar System or collision with the Sun or a planet are often estimated by Monte Carlo codes based on the equations of Öpik and using a method implemented by Arnold. Such algorithms assume that orbital changes are dominated by close encounters, and that successive encounters are uncorrelated. We have compared the results of an Öpik code (H. J. Melosh and W. B. Tonks, Mete-oritics 28, 398 (1993)) and a fast integrator (H. F. Levison and M. J. Duncan, Icarus 108, 18 (1994)) to investigate the regimes of validity of the Öpik–Arnold approach. We investigate the transfer of ecliptic comets from Neptune-crossing orbits to observable Jupiter-family comets, the dynamics of Halley-type comets, and the transport of meteorites among the terrestrial planets. In all cases, the Öpik code overestimates the median lifetime of the small bodies, although both codes show a rapid initial loss of objects followed by a slow decay. For martian impact ejecta, some of which find their way to Earth as the SNC meteorites, the Öpik code substantially overestimates lifetimes because of its neglect of secular resonances, which rapidly pump eccentricities (B. J. Gladman et al., Science 271, 1387 (1996)).