  We investigate how the ageing of stellar populations can drive a
morphological transformation in galaxies whose star formation (SF) activity has
been quenched on short timescales, like in cluster galaxies subject to ram
pressure stripping from the intracluster medium. For this purpose, we use a
sample of 91 galaxies with MUSE data from the GASP program and of their
spatially resolved SF history derived with the spectral modelling software
SINOPSIS. We simulate the future continuation of the SF activities by exploring
two quenching scenarios: an instantaneous truncation of the SF across the whole
disc, and an outside-in quenching with typical stripping timescales of 0.5 Gyr
and 1 Gyr. For each scenario we produce mock MUSE spectroscopic datacubes and
optical images for our galaxies during their evolution, and classify their
morphology using a new diagnostic tool, calibrated on cluster galaxies from the
OmegaWINGS Survey. We find that, in all scenarios considered, the initial
galaxy population dominated by blue-cloud spirals (90%) evolves into a mixed
population mostly composed by red-sequence spirals (50-55%) and lenticulars
(~40%). The morphology transformation is completed after just 1.5-3.5 Gyr,
proceeding faster in more efficient quenching scenarios. Our results indicate
that, even without accounting for dynamical processes, SF quenching caused by
the harsh environment of a cluster can significantly affect the morphology of
the infalling galaxy population on timescales of a few Gyr.
