  Globular clusters (GCs) are the ideal environment for the formation of
neutron stars (NSs) and millisecond pulsars (MSPs). NSs origin and evolution
provide a useful information on stellar dynamics and evolution in star
clusters, and are among the most interesting astrophysical objects, being
precursors of several high-energy phenomena such as gravitational waves and
gamma-ray bursts. Due to a large velocity kick that they receive at birth, most
of the NSs escape the local field, affecting the evolution and dynamics of
their parent cluster. In this paper, we study the origin and dynamical
evolution of NSs within GCs with different initial masses, metallicities and
primordial binary fractions. We find that the radial profile of NSs is shaped
by the BH content of the cluster, which partially quenches the NS segregation
until most of the BHs are ejected from the system. Independently on the cluster
mass and initial configuration, the NSs map the average stellar population, as
their average radial distance is $\approx 60-80\%$ of the cluster half-mass
radius. Finally, by assuming a recycling fraction of $f_\mathrm{rec}=0.1$ and
an average MSP gamma-ray emission of $L_\gamma=2\times 10^{33}$ erg s$^{-1}$,
we show that the typical gamma-ray emission from our GCs agrees with
observations and supports the MSP origin of the gamma-ray excess signal
observed by the Fermi-LAT telescope in the Galactic Centre.
