  We study the low-temperature transport properties of out-of-equilibrium XXZ
spin-$1/2$ chains. We consider the protocol where two semi-infinite chains are
prepared in two thermal states at small but different temperatures and suddenly
joined together. We focus on the qualitative and quantitative features of the
profiles of local observables, which at large times $t$ and distances $x$ from
the junction become functions of the ratio $\zeta=x/t$. By means of the
generalized hydrodynamic equations, we analyse the rich phenomenology arising
by considering different regimes of the phase diagram. In the gapped phases,
variations of the profiles are found to be exponentially small in the
temperatures but described by non-trivial functions of $\zeta$. We provide
analytical formulae for the latter, which give accurate results also for small
but finite temperatures. In the gapless regime, we show how the three-step
conformal predictions for the profiles of energy density and energy current are
naturally recovered from the hydrodynamic equations. Moreover, we also recover
the recent non-linear Luttinger liquid predictions for low-temperature
transport: universal peaks of width $\Delta\zeta\propto T$ emerge at the edges
of the light cone in the profiles of generic observables. Such peaks are
described by the same function of $\zeta$ for all local observables.
