  We study Euler scale hydrodynamics of massless integrable quantum field
theories interpolating between two non-trivial renormalisation group fixed
points after inhomogeneous quantum quenches. Using a partitioning protocol with
left and right initial thermal states and the recently developed framework of
generalised hydrodynamics, we focus on current and density profiles for the
energy and momentum as a function of \xi = x=t, where both x and t are sent to
infinity. Studying the first few members of the An and Dn massless flows we
carry out a systematic treatment of these series and generalise our results to
other unitary massless models. In our analysis we find that the profiles
exhibit extended plateaux and that non-trivial bounds exist for the energy and
momentum densities and currents in the non-equilibrium stationary state, i.e.
when \xi = 0. To quantify the magnitude of currents and densities, dynamical
central charges are defined and it is shown that the dynamical central charge
for the energy current satisfies a certain monotonicity property. We discuss
the connection of the Landauer-Buttiker formalism of transport with our results
and show that this picture can account for some of the bounds for the currents
and for the monotonicity of the dynamical central charge. These properties are
shown to be present not only in massless flows but also in the massive
sinh-Gordon model suggesting their general validity and the correctness of the
Landauer-Buttiker interpretation of transport in integrable field theories. Our
results thus imply the existence of a non-equilibrium c-theorem as well, at
least in integrable models. Finally we also study the interesting low energy
behaviour of the A2 model that corresponds to the massless flow from the
tricritical to the critical Ising field theory.
