The process is carried out at a pedestrian level in a way that spans the divide between relativistic and non-relativistic theory, even throwing in the "Carrollian" universe (called therein: the "Archimedean" case).
A Unified Framework For The Symplectic Wigner Classification
https://fdocuments.net/document/a-unified-framework-for-symplectic-wigner-classification.html
This could be generalized to cover all the kinematic groups in the Bacry Levi-Leblond (BLL) classification - in a uniform manner. The reason for the uniformity is that the Poisson manifolds associated with each of the Lie groups in this classification can all be combined into a single Poisson manifold. The BLL classification is, in fact, a three-parameter family of deformations of the Static Group. So, the corresponding unified Poisson manifold is obtained by just throwing in the parameters as three extra coordinates. So, the classification consists of the symplectic leaves of the combined Poisson manifold.
(This is in contrast to what happens if you create the classification by starting with the Static Group and finding all of its deformations. The resulting family is quite a bit larger than the BLL family, and not uniform at all.)
Thus, you can speak of continuously transitioning between the different kinematic groups. This supersedes the method of Lie Group contractions, and the results obtained therein.
A key point worth making here: uniformity can only be achieved after centrally extending all the kinematic groups. For the Galilei group, this yields the Bargmann group, which is more correctly regarded as the symmetry group for non-relativistic theory. What this deforms into is not the Poincaré group, but its (trivial) central extension. All the members in the BLL family have trivial central extensions, except those that have absolute simultaneity (that includes: Galilei, Static and the two Newton-Hooke groups). Both the Static and Carroll (or "Archimedean") group have the same central extension, even though the central extension is non-trivial for the Static group.
This has a material bearing on the classification question for Relativity as follows: our current understanding of the Bargmann group being the correct symmetry group for non-relativistic theory arose only late in the game, some time in the 1950's, well past the time that non-relativistic theory was superseded by relativity. So, it represents a retro-update to Newtonian physics. Before that, it was the Galilei group that was understood to be the symmetry group for non-relativistic theory.
But Relativity, itself, was formulated with a Correspondence Principle that connected to the older understanding of non-relativistic theory. The retro-update of the latter entails a revision of the Correspondence Principle, too: it should now retarget a theory that has been lifted, but the other end of the "Correspondence Principle" arrow, on the Relativity side, has not been lifted to match this retargeting. It also needs to be lifted. The result adds in an 11th generator, for the trivial central extension. Doing this also happens to clarify other seemingly obscure issues, like why the Dirac algebra needs to be complexified or why/how the "zero of energy is relative" (as is often asserted in the setting of quantum field theory, but is not justified from the vantage point of Poincaré group representations).
The resulting classification is actually the same as it is for the Poincaré group, but an extra degree of freedom appears in each subclass, arising from the extra generator. For the vacuum class, the extra degree of freedom corresponds to "vacuum energy".
Moreover, BLL family, in its entirety, admits a uniform (but generally non-linear) five-dimensional coordinate representation that includes the Bargmann geometry of non-relativistic theory as one of its cases. The centrally extended Poincaré group is associated with the geometry that has $dx^2 + dy^2 + dz^2 + 2 dt du + (1/c)^2 du^2$ and $ds = dt + (1/c)^2 du$ as its invariants, the latter invariant being the differential for proper time $s$, itself. Minkowski geometry is obtained by setting the former invariant to 0 (i.e. as a light cone in a 4+1 dimensional geometry). For non-relativistic theory, the corresponding geometry is obtained by replacing each of the $(1/c)^2$ factors by 0.
Time permitting, it might get written up and updated to a form suitable for for JMP. But, it's not a high-priority issue. But it probably should be TeX'ed and cleaned up and submitted, though I prefer an IoP journal.