Macdonald processes are certain probability measures on two-dimensional arrays of interlacing particles introduced by Borodin and Corwin in 7. They are defined in terms of nonnegative specializations of the Macdonald symmetric functions and depend on two parameters q,t∈0;1). Our main result is a classification of continuous time, nearest neighbor Markov dynamics on the space of interlacing arrays that act nicely on Macdonald processes. The classification unites known examples of such dynamics and also yields many new ones. When t=0, one dynamics leads to a new integrable interacting particle system on the one-dimensional lattice, which is a q-deformation of the PushTASEP (= long-range TASEP). When q=t, the Macdonald processes become the Schur processes of Okounkov and Reshetikhin 41. In this degeneration, we discover new Robinson–Schensted-type correspondences between words and pairs of Young tableaux that govern some of our dynamics.