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Iterating the skew RSK correspondence discovered by Sagan and Stanley in thelate '80s, we define a deterministic dynamics on the space of pairs of skewYoung tableaux $(P,Q)$. We find that this skew RSK dynamics displaysconservation laws which, in the picture of Viennot's shadow line construction,identify generalizations of Greene invariants.  The introduction of a novel realization of $0$-th Kashiwara operators revealsthat the skew RSK dynamics possesses symmetries induced by an affine bicrystalstructure,  which, combined with connectedness properties of Demazure crystals, leads toits linearization.  Studying asymptotic evolution of the dynamics started from a pair of skewtableaux $(P,Q)$, we discover a new bijection $\Upsilon : (P,Q) \mapsto (V,W;\kappa, \nu)$. Here $(V,W)$ is a pair of  vertically strict tableaux, i.e., column strict filling of Young diagramswith no condition on rows, with shape prescribed by the Greene invariant,$\kappa$ is an array of non-negative weights and $\nu$ is a partition.  An application of this construction is the first bijective proof of Cauchyand Littlewood identities involving $q$-Whittaker polynomials. New identitiesrelating sums of $q$-Whittaker and Schur polynomials are also presented.

Skew RSK dynamics: Greene invariants, affine crystals and applications to $q$-Whittaker polynomials