We revisit early suggestions to observe spin-charge separation (SCS) in
cold-atom settings {in the time domain} by studying one-dimensional repulsive
Fermi gases in a harmonic potential, where pulse perturbations are initially
created at the center of the trap. We analyze the subsequent evolution using
generalized hydrodynamics (GHD), which provides an exact description, at large
space-time scales, for arbitrary temperature $T$, particle density, and
interactions. At $T=0$ and vanishing magnetic field, we find that, after a
nontrivial transient regime, spin and charge dynamically decouple up to
perturbatively small corrections which we quantify. In this limit, our results
can be understood based on a simple phase-space hydrodynamic picture. At finite
temperature, we solve numerically the GHD equations, showing that for low $T>0$
effects of SCS survive and {characterize} explicitly the value of $T$ for which
the two distinguishable excitations melt.
