While wide-field surveys of M31 have revealed much substructure at large radii, understanding the nature and origin of this material is not straightforward from morphology alone. Using deep Hubble Space Telescope/Advanced Camera for Surveys data, we have derived further constraints in the form of quantitative star formation histories (SFHs) for 14 inner halo fields which sample diverse substructures. In agreement with our previous analysis of colour-magnitude diagram morphologies, we find the resultant behaviours can be broadly separated into two categories. The SFHs of `disc-like' fields indicate that most of their mass has formed since z ˜ 1, with one quarter of the mass formed in the last 5 Gyr. We find `stream-like' fields to be on average 1.5 Gyr older, with ≲ 10 per cent of their stellar mass formed within the last 5 Gyr. These fields are also characterized by an age-metallicity relation showing rapid chemical enrichment to solar metallicity by z = 1, suggestive of an early-type progenitor. We confirm a significant burst of star formation 2 Gyr ago, discovered in our previous work, in all the fields studied here. The presence of these young stars in our most remote fields suggests that they have not formed in situ but have been kicked-out from the thin disc through disc heating in the recent past.