We study inhomogeneous quenches in integrable models. The Non-Equilibrium Steady State emerging in such systems has been recently conjectured to be described by a Generalised Hydrodynamic theory. We develop a mathematically rigorous method to calculate the asymptotics of observables at large times and distances and show how certain predictions of this conjecture can be derived from analyticity properties of the Slavnov formula for Bethe state overlaps.