  We present comprehensive multi-component dynamical models of M54 (NGC6715),
the nuclear star cluster of the Sagittarius dwarf galaxy (Sgr), which is
undergoing a tidal disruption in the Milky Way halo. Previous papers in the
series used a large MUSE mosaic data set to identify multiple stellar
populations in the system and study their kinematic differences. Here we use
Jeans-based dynamical models that fit the population properties (mean age and
metallicity), spatial distributions, and kinematics simultaneously. They
provide a solid physical explanation to our previous findings. The
population-dynamical models deliver a comprehensive view of the whole system,
and allow us to disentangle the different stellar populations. We explore their
dynamical interplay and confirm our previous findings about the build-up of
Sgr's nuclear cluster via contributions from globular cluster stars, Sgr inner
field stars, and in-situ star formation. We explore various parameterisations
of the gravitational potential and show the importance of a radially varying
mass-to-light ratio for the proper treatment of the mass profile. We find a
total dynamical mass within M54's tidal radius ($\sim75$ pc) of
$1.60\pm0.07\times10^6$ Msun in excellent agreement with $N$-body simulations.
The metal-poor globular cluster stars contribute about $65\%$ of the total mass
or $1.04\pm0.05\times10^6$ Msun. The metal-rich stars can be further divided
into young and intermediate age populations that contribute
$0.32\pm0.02\times10^6$ Msun ($20\%$) and $0.24\pm0.02\times10^6$ Msun
($15\%$), respectively. Our population-dynamical models successfully
distinguish the different stellar populations in Sgr's nucleus because of their
different spatial distributions, ages, metallicities, and kinematic features.
