  The derivation of determinant representations for the space-, time-, and
temperature-dependent correlation functions of the impenetrable Gaudin-Yang
model in the presence of a trapping potential is presented. These
representations are valid in both equilibrium and nonequilibrium scenarios like
the ones initiated by a sudden change of the confinement potential. In the
equal-time case our results are shown to be equivalent to a multicomponent
generalization of Lenard's formula from which Painlev\'e transcendent
representations for the correlators can be obtained in the case of harmonic
trapping and Dirichlet and Neumann boundary conditions. For a system in the
quantum Newton's cradle setup the determinant representations allow for an
exact numerical investigation of the dynamics and even hydrodynamization which
is outside the reach of Generalized Hydrodynamics or other approximate methods.
In the case of a sudden change in the trap's frequency we predict a many-body
bounce effect, not present in the evolution of the density profile, which
causes a nontrivial periodic narrowing of the momentum distribution with
amplitude depending on the statistics of the particles.
