Bars are a major driver of secular evolution in disc galaxies, promoting the inflow of gas to the centre, where stellar structures, such as nuclear discs are built. We constrain the formation of these structures by deriving their stellar kinematics and mean population properties. To this end, we use observations with unprecedented spatial resolution, obtained with the MUSE integral-field spectrograph for a sample of 21 Milky Way-type galaxies in the local Universe. We show that nuclear discs are characterised by a high rotational support, i.e. near-circular orbits with low velocity dispersions, and are significantly younger, more metal-rich, and less [α/Fe]-enhanced, as compared to their surroundings. These findings are consistent with the picture of bar-driven secular evolution and contrast with the formation of old and kinematically hot classical bulges in violent accretion events. Moreover, nuclear discs exhibit well-defined radial gradients of the population properties with single slopes, suggesting that they are continuous components from their outer edge to the galaxy centre. We argue that these continuous (stellar) nuclear discs may form from a series of bar-built (gas-rich) nuclear rings that grow in radius, as the bar evolves. In this picture, nuclear rings are simply the star-forming outer edge of nuclear discs. Finally, we do not find evidence for the presence of classical bulges in the centres of these galaxies. This could result from feedback processes efficiently preventing the formation of classical bulges or may challenge the paradigm of hierarchical structure formation, questions we will address in a dedicated MUSE survey.