Results. We compared the Gaia Data Release 2 and Gaia EDR3 performances in the study of the Magellanic Clouds and show the clear improvements in precision and accuracy in the new release. We also show that the systematics still present in the data make the determination of the 3D geometry of the LMC a difficult endeavour; this is at the very limit of the usefulness of the Gaia EDR3 astrometry, but it may become feasible with the use of additional external data. We derive radial and tangential velocity maps and global profiles for the LMC for the several subsamples we defined. To our knowledge, this is the first time that the two planar components of the ordered and random motions are derived for multiple stellar evolutionary phases in a galactic disc outside the Milky Way, showing the differences between younger and older phases. We also analyse the spatial structure and motions in the central region, the bar, and the disc, providing new insightsinto features and kinematics. Finally, we show that the Gaia EDR3 data allows clearly resolving the Magellanic Bridge, and we trace the density and velocity flow of the stars from the SMC towards the LMC not only globally, but also separately for young and evolved populations. This allows us to confirm an evolved population in the Bridge that is slightly shift from the younger population. Additionally, we were able to study the outskirts of both Magellanic Clouds, in which we detected some well-known features and indications of new ones.