We introduce and study q-randomized Robinson-Schensted-Knuth (RSK) correspondences which interpolate between the classical (q=0) and geometric (q->1) RSK correspondences (the latter ones are sometimes also called tropical). For 0<q<1 our correspondences are randomized, i.e., the result of an insertion is a certain probability distribution on semistandard Young tableaux. Because of this randomness, we use the language of discrete time Markov dynamics on two-dimensional interlacing particle arrays (these arrays are in a natural bijection with semistandard tableaux). Our dynamics act nicely on a certain class of probability measures on arrays, namely, on q-Whittaker processes (which are t=0 versions of Macdonald processes). We present four Markov dynamics which for q=0 reduce to the classical row or column RSK correspondences applied to a random input matrix with independent geometric or Bernoulli entries. Our new two-dimensional discrete time dynamics generalize and extend several known constructions: (1) The discrete time q-TASEPs arise as one-dimensional marginals of our "column" dynamics. In a similar way, our "row" dynamics lead to discrete time q-PushTASEPs - new integrable particle systems in the Kardar-Parisi-Zhang universality class. We employ these new one-dimensional discrete time systems to establish a Fredholm determinantal formula for the two-sided continuous time q-PushASEP conjectured by Corwin-Petrov (2013). (2) In a certain Poisson-type limit (from discrete to continuous time), our two-dimensional dynamics reduce to the q-randomized column and row Robinson-Schensted correspondences introduced by O'Connell-Pei (2012) and Borodin-Petrov (2013), respectively. (3) In a scaling limit as q->1, two of our four dynamics on interlacing arrays turn into the geometric RSK correspondences associated with log-Gamma or strict-weak directed random polymers.