Context.The Andromeda (M 31) galaxy displays several substructures in its inner halo. Different simulations associate their origin with either a single relatively massive merger, or with a larger number of distinct, less massive accretions.Aims.The origin of these substructures as remnants of accreted satellites or perturbations of the pre-existing disc would be encoded in the properties of their stellar populations (SPs). The metallicity and star formation history of these distinct populations leave traces on their deep [O III] 5007 Å planetary nebulae luminosity function (PNLF). By characterizing the morphology of the PNLFs, we constrain their origin.Methods.From our 54 sq. deg. deep narrow-band [O III] survey of M 31, we identify planetary nebulae in six major inner-halo substructures: the Giant Stream, North East Shelf, G1 Clump, Northern Clump, Western Shelf, and Stream D. We obtain their PNLFs and those in two disc annuli, with galactocentric radii ofR_GC= 10–20 kpc andR_GC= 20–30 kpc. We measure PNLF parameters from cumulative fits and statistically compare the PNLFs in each substructure and disc annulus. We link these deep PNLF parameters and those for the Large Magellanic Cloud (LMC) to published metallicities and resolved stellar population-age measurements for their parent SPs.Results.The absolute magnitudes (M^*) of the PNLF bright cut-off for these sub-populations span a significant magnitude range, despite being located at the same distance and having a similar line-of-sight extinction. TheM^*values of the Giant Stream, W Shelf, and Stream D PNLFs are fainter than those predicted by PN evolution models by 0.6, 0.8, and 1.5 mag, respectively, assuming the measured metallicity of the parent stellar populations. The faint-end slope of the PNLF increases linearly with decreasing fraction of stellar mass younger than 5 Gyr across the M 31 regions and the LMC. From their PNLFs, the Giant Stream and NE Shelf are consistent with being stellar debris from an infalling satellite, while the G1 Clump appears to be linked with the pre-merger disc with an additional contribution from younger stars.Conclusions.The SPs of the substructures are consistent with those predicted by simulations of a single fairly massive merger event that took place 2–3 Gyr ago in M31. Stream D has an unrelated, distinct origin. Furthermore, this study provides independent evidence that the faint-end of the PNLF is preferentially populated by planetary nebulae evolved from older stars.