Context. The motion of stars has been used to reveal details of the complex history of the Milky Way, in constant interaction with its environment. Nevertheless, to reconstruct the Galactic history puzzle in its entirety, the chemo-physical characterisation of stars is essential. Previous Gaia data releases were supported by a smaller, heterogeneous, and spatially biased mixture of chemical data from ground-based observations. Aims. Gaia Data Release 3 opens a new era of all-sky spectral analysis of stellar populations thanks to the nearly 5.6 million stars observed by the Radial Velocity Spectrometer (RVS) and parametrised by the GSP-Spec module. In this work, we aim to demonstrate the scientific quality of Gaia s Milky Way chemical cartography through a chemo-dynamical analysis of disc and halo populations. Methods. Stellar atmospheric parameters and chemical abundances provided by Gaia DR3 spectroscopy are combined with DR3 radial velocities and EDR3 astrometry to analyse the relationships between chemistry and Milky Way structure, stellar kinematics, and orbital parameters. Results. The all-sky Gaia chemical cartography allows a powerful and precise chemo-dynamical view of the Milky Way with unprecedented spatial coverage and statistical robustness. First, it reveals the strong vertical symmetry of the Galaxy and the flared structure of the disc. Second, the observed kinematic disturbances of the disc seen as phase space correlations and kinematic or orbital substructures are associated with chemical patterns that favour stars with enhanced metallicities and lower [α/Fe] abundance ratios compared to the median values in the radial distributions. This is detected both for young objects that trace the spiral arms and older populations. Several α, iron-peak elements and at least one heavy element trace the thin and thick disc properties in the solar cylinder. Third, young disc stars show a recent chemical impoverishment in several elements. Fourth, the largest chemo-dynamical sample of open clusters analysed so far shows a steepening of the radial metallicity gradient with age, which is also observed in the young field population. Finally, the Gaia chemical data have the required coverage and precision to unveil galaxy accretion debris and heated disc stars on halo orbits through their [α/Fe] ratio, and to allow the study of the chemo-dynamical properties of globular clusters. Conclusions. Gaia DR3 chemo-dynamical diagnostics open new horizons before the era of ground-based wide-field spectroscopic surveys. They unveil a complex Milky Way that is the outcome of an eventful evolution, shaping it to the present day.