The chemical evolution of neutron capture elements in the Milky Way disc is still a matter of debate. There is a lack of statistically significant catalogues of such element abundances, especially those of the r-process. We aim to understand the chemical evolution of r-process elements in Milky Way disc. We focus on three pure r-process elements Eu, Gd, and Dy. We also consider a pure s-process element, Ba, in order to disentangle the different nucleosynthesis processes. We take advantage of high-resolution FEROS, HARPS, and UVES spectra from the ESO archive in order to perform a homogeneous analysis on 6500 FGK Milky Way stars. The chemical analysis is performed thanks to the automatic optimization pipeline GAUGUIN. We present abundances of Ba (5057 stars), Eu (6268 stars), Gd (5431 stars), and Dy (5479 stars). Based on the [{alpha}/Fe] ratio determined previously by the AMBRE Project, we chemically characterize the thin and the thick discs, and a metal-rich {alpha}-rich population. First, we find that the [Eu/Fe] ratio follows a continuous sequence from the thin disc to the thick disc as a function of the metallicity. Second, in thick disc stars, the [Eu/Ba] ratio is found to be constant, while the [Gd/Ba] and [Dy/Ba] ratios decrease as a function of the metallicity. These observations clearly indicate a different nucleosynthesis history in the thick disc between Eu and Gd-Dy. The [r/Fe] ratio in the thin disc is roughly around +0.1dex at solar metallicity, which is not the case for Ba. We also find that the {alpha}-rich metal-rich stars are also enriched in r-process elements (like thick disc stars), but their [Ba/Fe] is very different from thick disc stars. Finally, we find that the [r/{alpha}] ratio tends to decrease with metallicity, indicating that supernovae of different properties probably contribute differently to the synthesis of r-process elements and {alpha}-elements. We provide average abundance trends for [Ba/Fe] and [Eu/Fe] with rather small dispersions, and for the first time for [Gd/Fe] and [Dy/Fe]. This data may help to constrain chemical evolution models of Milky Way r- and s-process elements and the yields of massive stars. We emphasize that including yields of neutron-star or black hole mergers is now crucial if we want to quantitatively compare observations to Galactic chemical evolution models.