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 red giant branch and the AGB, in agreement with previous studies. For NGC 1978 it is possible to explain the C/O and C-12/C-13 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.