  Benefiting from the GAIA second and early third releases of photometric and
astrometric data we examine the population of asymptotic giant branch (AGB)
stars that appear in the fields of intermediate-age and young open star
clusters. We identify 49 AGB star candidates, brighter than the tip of the red
giant branch, with a good-to-high cluster membership probability. Among them we
find 19 TP-AGB stars with known spectral type: 4 M stars, 3 MS/S stars and 12 C
stars. By combining observations, stellar models, and radiative transfer
calculations that include the effect of circumstellar dust, we characterize
each star in terms of initial mass, luminosity, mass-loss rate, core mass,
period and mode of pulsation. The information collected helps us shed light on
the TP-AGB evolution at solar-like metallicity, placing constraints on the
third dredge-up process, the initial masses of carbon stars, stellar winds, and
the initial-final mass relation (IFMR). In particular, we find that two bright
carbon stars, MSB 75 and BM IV 90, members of the clusters NGC 7789 and NGC
2660 (with similar ages of $\simeq 1.2-1.6$ Gyr and initial masses $ 2.1 \ge
M_{\rm i}/M_{\odot} \ge 1.9$), have unusually high core masses, $M_{\rm c}
\approx 0.67-0.7\,M_{\odot}$. These results support the findings of a recent
work (Marigo et al. 2020) that identified a kink in the IFMR, which interrupts
its monotonic trend just at the same initial masses. Finally, we investigate
two competing scenarios to explain the $M_{\rm c}$ data: the role of stellar
winds in single-star evolution, and binary interactions through the
blue-straggler channel.
