  We study the time evolution of an initially trapped weakly interacting Bose
gas at positive temperature, after the trapping potential has been switched
off. It has been recently shown in arXiv:2009.00992 that the one-particle
density matrix of Gibbs states of the interacting trapped gas is given, to
leading order in $N$, as $N \to \infty$, by the one of the ideal gas, with the
condensate wave function replaced by the minimizer of the Hartree energy
functional. We show that this structure is stable with respect to the many-body
evolution in the following sense: the dynamics can be approximated in terms of
the time-dependent Hartree equation for the condensate wave function and in
terms of the free evolution for the thermally excited particles. The main
technical novelty of our work is the use of the Hartree-Fock-Bogoliubov
equations to define a fluctuation dynamics.
