  Star clusters appear to be the ideal environment for the assembly of neutron
star-neutron star (NS-NS) and black hole-neutron star (BH-NS) binaries. These
binaries are among the most interesting astrophysical objects, being potential
sources of gravitational waves (GWs) and gamma-ray bursts. We use for the first
time high-precision N-body simulations of young massive and open clusters to
study the origin and dynamical evolution of NSs, within clusters with different
initial masses, metallicities, primordial binary fractions, and prescriptions
for the compact object natal kicks at birth. We find that the radial profile of
NSs is shaped by the BH content of the cluster, which partially quenches the NS
segregation due to the BH-burning process. This leaves most of the NSs out of
the densest cluster regions, where NS-NS and BH-NS binaries could potentially
form. Due to a large velocity kick that they receive at birth, most of the NSs
escape the host clusters, with the bulk of their retained population made up of
NSs of $\sim 1.3$ M$_\odot$ coming from the electron-capture supernova process.
The details of the primordial binary fraction and pairing can smear out this
trend. Finally, we find that a subset of our models produce NS-NS mergers,
leading to a rate of $\sim 0.01$--$0.1$ Gpc$^{-3}$ yr$^{-1}$ in the local
Universe, and compute an upper limit of $\sim 3\times 10^{-2}$--$3\times
10^{-3}$ Gpc$^{-3}$ yr$^{-1}$ for the BH-NS merger rate. Our estimates are
several orders of magnitude smaller than the current empirical merger rate from
LIGO/Virgo, in agreement with the recent rate estimates for old globular
clusters.
