  Elastic collisions with relativistic electrons from the blazar's jet can
accelerate dark matter (DM) particles in the DM spike surrounding the
supermassive black hole at its center. This can allow one to set stringent
limits on the DM-electron scattering cross section ($\bar{\sigma}_{e\chi}$) for
DM masses less than 100 MeV. We consider DM particles boosted by energetic
electrons in the jets of the blazars TXS 0506+056 and BL Lacertae. Both vector
and scalar mediators for the scattering of electron and electrophilic fermionic
DM are studied. We highlight that the ensuing energy dependency of the S-matrix
for the corresponding Lorentz structure of the vertex significantly modifies
the constraints. We find that the revised exclusion limits are orders of
magnitude stronger than the equivalent results for the simple constant cross
section assumption. Our limits are also assessed for the less cuspy spike.
