  Macdonald processes are certain probability measures on two-dimensional
arrays of interlacing particles introduced by Borodin and Corwin
(arXiv:1111.4408 [math.PR]). They are defined in terms of nonnegative
specializations of the Macdonald symmetric functions and depend on two
parameters (q,t), where 0<= q, t < 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 (arXiv:math/0107056 [math.CO]). In
this degeneration, we discover new Robinson--Schensted-type correspondences
between words and pairs of Young tableaux that govern some of our dynamics.
