The Andromeda (M31) galaxy displays several substructures in its inner halo whose origin as remnants of accreted satellites or perturbations of the pre-existing disc are encoded in the properties of their stellar populations (SPs), leaving traces on their deep [OIII] 5007 AA planetary nebulae luminosity functions (PNLFs). By characterizing the morphology of the PNLFs, we constrain their origin. From our 54 sq. deg. deep narrow-band [OIII] survey of M31, we identify planetary nebulae (PNe) in the M31 disc and six major inner-halo substructures - the Giant Stream, North East Shelf, G1-Clump, Northern Clump, Western Shelf and Stream-D. We measure PNLF parameters from cumulative fits and statistically compare the PNLFs in each substructure and the disc. We link the PNLF parameters and those for the Large Magellanic Cloud to published metallicities and age measurements for their parent SPs. The absolute magnitudes of the PNLF bright cut-off ($M^{*}$) for these sub-populations span a significant magnitude range, despite having similar distance and line-of-sight extinction. $M^{*}$ for the Giant Stream, W-shelf and Stream-D PNLFs are fainter than those predicted by PN evolution models for the metallicity of the parent SPs. The faint-end slope of the PNLF increases linearly with decreasing fraction of stellar mass younger than 5 Gyr across the M31 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. The SPs of the substructures are consistent with those predicted by simulations of a single 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 PNe evolved from older stars.
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