  (Abridged) We demonstrate that the amount of extra mixing required to fit the
observed low C/N and 12C/13C ratios in first giant branch (FGB) stars is also
sufficient to explain the C and N abundances of Galactic AGB stars. We simulate
the effect of extra mixing on the FGB by setting the composition of the
envelope to that observed in low-mass FGB stars, and then evolve the models to
the tip of the AGB. The inclusion of FGB extra mixing compositional changes has
a strong effect on the C and N abundance in our AGB models, leading to
compositions consistent with those measured in Galactic C-rich stars. The
composition of the models is also consistent with C abundances measured in
mainstream silicon carbide grains. While our models cover the range of C
abundances measured in C stars in NGC 1846, we cannot simultaneously match the
composition of the O and C-rich stars. Our models only match the O isotopic
composition of K and some M, MS giants, and are not able to match the O
composition of C-rich AGB stars. By increasing the 16O intershell abundance
(based on observational evidence) it is possible to reproduce the observed
trend of increasing 16O/18O and 16O/17O ratios with evolutionary phase. We
conclude 1) if extra mixing occurs during the AGB it likely only occurs
efficiently in low metallicity objects, or when the stars are heavily obscured
making spectroscopic observations difficult, and 2) that the intershell
compositions of AGB stars needs further investigation.
