  Kuiper belt objects, such as Arrokoth, the probable progenitors of
short-period comets, formed and evolved at large heliocentric distances, where
the ambient temperatures appear to be sufficiently low for preserving volatile
ices. By detailed numerical simulations, we follow the long-term evolution of
small bodies, composed of amorphous water ice, dust, and ices of other volatile
species that are commonly observed in comets. The heat sources are solar
radiation and the decay of short-lived radionuclides. The bodies are highly
porous and gases released in the interior flow through the porous medium. The
most volatile ices, CO and CH$_4$ , are found to be depleted down to the center
over a time scale on the order of 100 Myr. Sublimation fronts advance from the
surface inward, and when the temperature in the inner part rises sufficiently,
bulk sublimation throughout the interior reduces gradually the volatile ices
content until they are completely lost. All the other ices survive, which is
compatible with data collected by New Horizons on Arrokoth, showing the
presence of methanol, and possibly, H$_2$O, CO$_2$, NH$_3$ and C$_2$H$_6$, but
no hypervolatiles. The effect of short-lived radionuclides is to increase the
sublimation equilibrium temperatures and reduce volatile depletion times. We
consider the effect of the bulk density, abundance ratios and heliocentric
distance. At 100~au, CO is depleted, but CH$_4$ survives to present time,
except for a thin outer layer. Since CO is abundantly detected in comets, we
conclude that the source of highly volatile species in active comets must be
gas trapped in amorphous ice.
