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Aims: We investigate the change in the surface abundance of $^{12}$C duringthe evolution along the AGB, aiming to constrain third dredge-up models.  Methods: High-resolution, near-infrared spectra of a sample of AGB stars inthe LMC cluster NGC 1846 were obtained. A cluster sample ensures a high levelof homogeneity with respect to age, metallicity, and distance. The C/O ratioand the ratio of $^{12}$C/$^{13}$C were measured and compared with ourevolutionary models.  Results: For the first time, we show the evolution of the C/O and$^{12}$C/$^{13}$C ratios along a cluster AGB. Our findings allow us to checkthe reliability of the evolutionary models and, in particular, the efficiencyof the third dredge up. The increase in both C/O and $^{12}$C/$^{13}$C in theobserved O-rich stars is reproduced by the models well. However, the low carbonisotopic ratios of the two C-stars in our sample indicate the late occurrenceof moderate extra mixing. The extra mixing affects the most luminous AGB starsand is capable of increasing the abundance of $^{13}$C, while leaving unchangedthe C/O ratio, which has been fixed by the cumulative action of several thirddredge-up episodes. We find indications that the F abundance also increasesalong the AGB, supporting an in situ production of this element.

AGB stars of the intermediate-age LMC cluster NGC 1846 II. Dredge up along the AGB