It remains an open question as to how long ago the morphology that we see in a present-day galaxy was typically imprinted. Studies of galaxy populations at different redshifts reveal that the balance of morphologies has changed over time, but such snapshots cannot uncover the typical timescales over which individual galaxies undergo morphological transformation, nor which are the progenitors of todays galaxies of different types. However, these studies also show a strong link between morphology and star-formation rate over a large range in redshift, which offers an alternative probe of morphological transformation. We therefore derive the evolution in star-formation rate and stellar mass of a sample of 4342 galaxies in the SDSS-IV MaNGA survey through a stellar population fossil record approach, and show that the average evolution of the population shows good agreement with known behaviour from previous studies. Although the correlation between a galaxys contemporaneous morphology and star-formation rate is strong over a large range of lookback times, we find that a galaxys present-day morphology only correlates with its relatively recent (~2 Gyr) star-formation history. We therefore find strong evidence that morphological transitions to galaxies current appearance occurred on timescales as short as a few billion years.