In the Gaia era, understanding the effects of the perturbations of the Galactic disc is of major importance in the context of dynamical modelling. In this theoretical paper we extend previous work in which, making use of the epicyclic approximation, the linearized Boltzmann equation had been used to explicitly compute, away from resonances, the perturbed distribution function of a Galactic thin-disc population in the presence of a non-axisymmetric perturbation of constant amplitude. Here we improve this theoretical framework in two distinct ways in the new code that we present. First, we use better estimates for the action-angle variables away from quasi-circular orbits, computed from the AGAMA software, and we present an efficient routine to numerically re-express any perturbing potential in these coordinates with a typical accuracy at the per cent level. The use of more accurate action estimates allows us to identify resonances such as the outer 1:1 bar resonance at higher azimuthal velocities than the outer Lindblad resonance, and to extend our previous theoretical results well above the Galactic plane, where we explicitly show how they differ from the epicyclic approximation. In particular, the displacement of resonances in velocity space as a function of height can in principle constrain the 3D structure of the Galactic potential. Second, we allow the perturbation to be time dependent, thereby allowing us to model the effect of transient spiral arms or a growing bar. The theoretical framework and tools presented here will be useful for a thorough analytical dynamical modelling of the complex velocity distribution of disc stars as measured by past and upcoming Gaia data releases.