  Galactic archaeology largely relies on precise ages of distant evolved stars
in the Milky Way. Nowadays, asteroseismology can deliver ages for many red
giants observed with high-cadence, high-precision photometric space missions.
Our aim is to quantify age uncertainties of slowly-rotating red giants due to
the cumulative effect of their fast rotation during core-hydrogen burning.
Their rotation in earlier evolutionary phases caused mixing resulting in
heavier helium cores and the prolongation of their main sequence. These
rotational effects are usually ignored when age-dating red giants, despite our
knowledge of fast rotation for stars with $M\ge1.3\,$M$_\odot$. We use a sample
of 490 $\gamma$ Doradus pulsators with precise asteroseismic estimates of their
internal rotation rate and with luminosity estimates from Gaia. For this
sample, which includes stars rotating from nearly 0 to about 60% of the
critical rate, we compute the cumulative effect on the age in their post-main
sequence evolution caused by rotational mixing on the main sequence. We use
stellar model grids with different physical prescriptions mimicking rotational
mixing to assess systematic uncertainties on the age. With respect to
non-rotating models, the sample of 490 stars, as red giant progenitors, reveals
age differences up to 5% by the time they start hydrogen-shell burning when
relying on the theory of rotationally induced diffusive mixing as included in
the MIST isochrones. Using rotational mixing based on an advective-diffusive
approach including meridional circulation leads to an age shift of 20% by the
time of the TRGB. Age-dating of red giants is affected by the cumulative effect
of rotational mixing during the main sequence. Such rotationally-induced age
shifts should be taken into account in addition to other effects if the aim is
to perform Galactic archaeological studies at the highest precision. (abridged)
