  The Sagittarius dwarf spheroidal galaxy (Sgr dSph) is in an advanced stage of
disruption but still hosts its nuclear star cluster (NSC), M54, at its center.
In this paper, we present a detailed kinematic characterization of the three
stellar populations present in M54: young metal-rich (YMR); intermediate-age
metal-rich (IMR); and old metal-poor (OMP), based on the spectra of $\sim6500$
individual M54 member stars extracted from a large MUSE/VLT dataset. We find
that the OMP population is slightly flattened with a low amount of rotation
($\sim0.8$ km s$^{-1}$) and with a velocity dispersion that follows a Plummer
profile. The YMR population displays a high amount of rotation ($\sim5$ km
s$^{-1}$) and a high degree of flattening, with a lower and flat velocity
dispersion profile. The IMR population shows a high but flat velocity
dispersion profile, with some degree of rotation ($\sim2$ km s$^{-1}$). We
complement our MUSE data with information from \textit{Gaia DR2} and confirm
that the stars from the OMP and YMR populations are comoving in 3D space,
suggesting that they are dynamically bound. While dynamical evolutionary
effects (e.g. energy equipartition) are able to explain the differences in
velocity dispersion between the stellar populations, the strong differences in
rotation indicate different formation paths for the populations, as supported
by an $N$-body simulation tailored to emulate the YMR-OMP system. This study
provides additional evidence for the M54 formation scenario proposed in our
previous work, where this NSC formed via GC accretion (OMP) and in situ
formation from gas accretion in a rotationally supported disc (YMR).
