In a hierarchically-formed Universe, the Milky Way is a test-bed to study in details the mechanisms that shape galaxies. The synergy between the Gaia space satellite and the ground-based spectroscopic survey WEAVE gives access, for the first time, to more than thirty tracers of the past of our Galaxy for a million stars of the extended Solar neighbourhood, and to a dozen of tracers for another two million stars outside of it. Our project concerns the study of the Galactic disc, a structure that encodes both internal (e.g. stellar radial migration) and external (e.g. accretion events) mechanisms that come into play in the chemo-dynamical evolution of our Galaxy. We have built a versatile team with nodes in Nice, Paris and Strasbourg, including experts in Galaxy evolution, simulations and modelling. The team members are heavily involved in both WEAVE and Gaia in order to extract the maximum of information available in those combined catalogues. Over the course of the four-year MWDisc ANR project, and alongside to the accumulation of the WEAVE data (starting in Q1 2021), we aim to produce added-value catalogues for the WEAVE stellar targets (containing homogeneous stellar chemical abundances, ages, orbits and extinctions) and models associated to the diffusion of mono-age populations by time-varying perturbations and superposition of perturbations (associated to the spiral arms and the Galactic bar). These models and catalogues will allow us, in turn, to evaluate the star formation history in various regions of the disc, put constraints on the merger tree of the Milky Way (including the analysis of existing simulations of ours), to link the geometrical properties of the thin and thick disc with their chemical counterparts and finally to characterize the efficiency of radial migration throughout the disc.