  The presence of AGB stars in clusters provides key constraints for stellar
models, as has been demonstrated with historical data from the Magellanic
Clouds. In this work, we look for candidate AGB stars in M31 star clusters from
the Panchromatic Hubble Andromeda Treasury (PHAT) survey. Our photometric
criteria selects stars brighter than the tip of the red giant branch, which
includes the bulk of the thermally-pulsing AGB stars as well as early-AGB stars
and other luminous cool giants expected in young stellar populations (e.g.
massive red supergiants, and intermediate-mass red helium-burning stars). The
AGB stars can be differentiated, a posteriori, using the ages already estimated
for our cluster sample. 937 candidates are found within the cluster aperture
radii, half (450) of which are very likely cluster members. Cross-matching with
additional databases reveals two carbon stars and ten secure variables among
them. The field-corrected age distribution reveals the presence of young
supergiants peaking at ages smaller than 100 Myr, followed by a long tail of
AGB stars extending up to the oldest possible ages. This long tail reveals the
general decrease in the numbers of AGB stars from initial values of 50e-6/Msun
at 100 Myr down to 5e-6/Msun at 10 Gyr. Theoretical models of near-solar
metallicity reproduce this general trend, although with localized discrepancies
over some age intervals, whose origin is not yet identified. The entire
catalogue is released together with finding charts to facilitate follow-up
studies.
