  Iterating the skew RSK correspondence discovered by Sagan and Stanley in the
late '80s, we define a deterministic dynamics on the space of pairs of skew
Young tableaux $(P,Q)$. We find that this skew RSK dynamics displays
conservation 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 reveals
that the skew RSK dynamics possesses symmetries induced by an affine bicrystal
structure,
  which, combined with connectedness properties of Demazure crystals, leads to
its linearization.
  Studying asymptotic evolution of the dynamics started from a pair of skew
tableaux $(P,Q)$, 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 diagrams
with 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 Cauchy
and Littlewood identities involving $q$-Whittaker polynomials. New identities
relating sums of $q$-Whittaker and Schur polynomials are also presented.
