  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.
