  Within gauge/gravity duality, we consider the local quench-like time
evolution obtained by joining two 1+1-dimensional heat baths at different
temperatures at time t=0. A steady state forms and expands in space. For the
2+1-dimensional gravity dual, we find that the shockwaves expanding the
steady-state region are of spacelike nature in the bulk despite being null at
the boundary. However, they do not transport information. Moreover, by adapting
the time-dependent Hubeny-Rangamani-Takayanagi prescription, we holographically
calculate the entanglement entropy and also the mutual information for
different entangling regions. For general temperatures, we find that the
entanglement entropy increase rate satisfies the same bound as in the
"entanglement tsunami" setups. For small temperatures of the two baths, we
derive an analytical formula for the time dependence of the entanglement
entropy. This replaces the entanglement tsunami-like behaviour seen for high
temperatures. Finally, we check that strong subadditivity holds in this
time-dependent system, as well as further more general entanglement
inequalities for five or more regions recently derived for the static case.
