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 time-scales over which individual galaxies undergo morphological transformation, nor which are the progenitors of today's galaxies of different types. However, these studies also show a strong link between morphology and star formation rate (SFR) over a large range in redshift, which offers an alternative probe of morphological transformation. We therefore derive the evolution in SFR 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 galaxy's contemporaneous morphology and SFR is strong over a large range of lookback times, we find that a galaxy's present-day morphology only correlates with its relatively recent (similar to 2 Gyr) star formation history. We therefore find strong evidence that morphological transitions to galaxies' current appearance occurred on time-scales as short as a few billion years.