  Ultra-luminous X-ray sources (ULXs) are those X-ray sources located away from
the centre of their host galaxy with luminosities exceeding the Eddington limit
of a stellar-mass black hole ($L_X>10^{39}\;{\rm erg\,s}^{-1}$). The discovery
of X-ray pulsations in some of these objects (e.g. M82~X-2) suggests that a
certain fraction of the ULX population may have a neutron star accretor. We
present systematic modelling of low- and intermediate-mass X-ray binaries
(LMXBs and IMXBs; donor-star mass range $0.92$--$8.0$~M$_{\odot}$ and
neutron-star accretors) to explain the formation of this sub-population of
ULXs. Using MESA, we explored the allowed initial parameter space of binary
systems consisting of a neutron star and a low- or intermediate-mass donor star
that could explain the observed properties of ULXs. Our simulations take into
account beaming effects, stellar rotation, general angular momentum losses, and
a detailed and self-consistent calculation of the mass-transfer rate. We study
the conditions that lead to dynamical stability of these systems, which depends
strongly on the response of the donor star to mass loss. Using two values for
the initial neutron star mass ($1.3$~M$_{\odot}$ and $2.0$~M$_{\odot}$), we
present two sets of mass-transfer calculation grids. We find that LMXBs/IMXBs
can produce NS-ULXs with typical time-averaged isotropic-equivalent X-ray
luminosities of $10^{39}$--$10^{41}\;{\rm erg\,s}^{-1}$ on a timescale up to
$\sim\!1.0\;{\rm Myr}$ for the lower luminosities. We also estimate their
likelihood of detection, the types of white-dwarf remnants left behind by the
donors, and the total amount of mass accreted by the neutron stars. We also
compare our results to the observed pulsating ULXs. Our results suggest that a
large subset of the observed pulsating ULX population can be explained by
LMXBs/IMXBs undergoing a super-Eddington mass-transfer phase.
