  The structure and dynamics of the central bar of the Milky Way are still
under debate whilst being fundamental ingredients for the evolution of our
Galaxy. The recent Gaia DR3 offers an unprecedented detailed view of the 6D
phase-space of the MW. We aim to characterise the dynamical moving groups
across the MW disc, and use their large-scale distribution to help constrain
the properties of the Galactic bar. We used wavelet transforms of the azimuthal
velocity ($V_\phi$) distribution in bins of radial velocity to robustly detect
the kinematic substructure in the Gaia DR3 catalogue. We then connected these
structures across the disc to measure the azimuthal ($\phi$) and radial ($R$)
gradients of the moving groups. We simulated thousands of perturbed
distribution functions using Backwards Integration of feasible Galaxy models
that include a bar, to compare them with the data and to explore and quantify
the degeneracies. The radial gradient of the Hercules moving group ($\partial
V_\phi/\partial R$ = 28.1$\pm$2.8 km$\,$s$^{-1}\,$kpc$^{-1}$) cannot be
reproduced by our simple models of the Galaxy which show much larger slopes
both for a fast and a slow bar. This suggests the need for more complex
dynamics (e.g. spiral arms, a slowing bar, external perturbations, etc.). We
measure an azimuthal gradient for Hercules of $\partial V_\phi/\partial \phi$ =
-0.63$\pm$0.13$\,$km$\,$s$^{-1}$deg$^{-1}$ and find that it is compatible with
both the slow and fast bar models. Our analysis points out that using this type
of analysis at least two moving groups are needed to start breaking the
degeneracies. We conclude that it is not sufficient for a model to replicate
the local velocity distribution; it must also capture its larger-scale
variations. The accurate quantification of the gradients, especially in the
azimuthal direction, will be key for the understanding of the dynamics
governing the disc. (ABR)
