We introduce and study four $q$-randomized Robinson--Schensted--Knuth (RSK) insertion tableau dynamics. Each of them is a discrete time Markov dynamics on two-dimensional interlacing particle arrays (these arrays are in a natural bijection with semistandard Young tableaux). For $0<q<1$ each dynamics provides a two-dimensional extension of the corresponding one-dimensional exactly solvable random dynamics of interacting particles. We prove that our dynamics act nicely on a certain class of probability measures on arrays, namely, on $q$-Whittaker processes. For $q=0$ these dynamics degenerate to the classical row or column RSK insertion tableau dynamics applied to a random input matrix with independent geometric or Bernoulli entries. We prove that in a scaling limit as $q\nearrow1$, two of our four dynamics on interlacing arrays turn into the geometric RSK dynamics associated with log-Gamma and strict-weak directed random lattice polymers.
