  We provide a new hydrodynamic framework to describe out-of-equilibrium
integrable systems with space-time inhomogeneous interactions. Our result
builds up on the recently-introduced Generalized Hydrodynamics (GHD). The
method allows to analytically describe the dynamics during generic
space-time-dependent smooth modulations of the interactions. As a proof of
concept, we study experimentally-motivated interaction quenches in the trapped
interacting Bose gas, which cannot be treated with current analytical or
numerical methods. We also benchmark our results in the XXZ spin chain and in
the classical Sinh-Gordon model.
