  We present stellar evolutionary sequences for asymptotic giant branch (AGB)
stars in the Magellanic Cloud clusters NGC 1978, NGC 1846 and NGC 419. The new
stellar models for the three clusters match the observed effective temperatures
on the giant branches, the oxygen-rich to carbon-rich transition luminosities,
and the AGB-tip luminosities. A major finding is that a large amount of
convective overshoot (up to 3 pressure scale heights) is required at the base
of the convective envelope during third dredge-up in order to get the correct
oxygen-rich to carbon-rich transition luminosity. The stellar evolution
sequences are used as input for detailed nucleosynthesis calculations. For NGC
1978 and NGC 1846 we compare our model results to the observationally derived
abundances of carbon and oxygen. We find that additional mixing processes
(extra-mixing) are required to explain the observed abundance patterns. For NGC
1846 we conclude that non-convective extra-mixing processes are required on
both the RGB and the AGB, in agreement with previous studies. For NGC 1978 it
is possible to explain the C/O and 12C/13C abundances of both the O-rich and
the C-rich AGB stars by assuming that the material in the intershell region
contains high abundances of both C and O. This may occur during a thermal pulse
when convective overshoot at the inner edge of the flash-driven convective
pocket dredges C and O from the core to the intershell. For NGC 419 we provide
our predicted model abundance values although there are currently no published
observed abundance studies for the AGB stars in this cluster.
