  Aims: We investigate the change in the surface abundance of $^{12}$C during
the evolution along the AGB, aiming to constrain third dredge-up models.
  Methods: High-resolution, near-infrared spectra of a sample of AGB stars in
the LMC cluster NGC 1846 were obtained. A cluster sample ensures a high level
of homogeneity with respect to age, metallicity, and distance. The C/O ratio
and the ratio of $^{12}$C/$^{13}$C were measured and compared with our
evolutionary 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 check
the reliability of the evolutionary models and, in particular, the efficiency
of the third dredge up. The increase in both C/O and $^{12}$C/$^{13}$C in the
observed O-rich stars is reproduced by the models well. However, the low carbon
isotopic ratios of the two C-stars in our sample indicate the late occurrence
of moderate extra mixing. The extra mixing affects the most luminous AGB stars
and is capable of increasing the abundance of $^{13}$C, while leaving unchanged
the C/O ratio, which has been fixed by the cumulative action of several third
dredge-up episodes. We find indications that the F abundance also increases
along the AGB, supporting an in situ production of this element.
