Unitary integrable models typically relax to a stationary generalized Gibbs ensemble (GGE), but in experimental realizations dissipation often breaks integrability. In this work, we use the recently introduced time-dependent GGE (t-GGE) approach to describe the open dynamics of a gas of bosons subject to atom losses and gains. We employ tensor network methods to provide numerical evidence of the exactness of the t-GGE in the limit of adiabatic dissipation, and of its accuracy in the regime of weak but finite dissipation. That accuracy is tested for two-point functions via the rapidity distribution, and for more complicated correlations through a non-Gaussianity measure. We combine this description with generalized hydrodynamics and we show that it correctly captures transport at large scales. Our results demonstrate that the t-GGE approach is robust in both homogeneous and inhomogeneous settings.
