Our journey starts from interacting random walks with push-and-block dynamics. We then consider their positive temperature analogues, touching upon polymer partition functions. Finally, we arrive at matrix Whittaker processes, which are integrable models of interacting Markov dynamics on matrix spaces. Our main tools are intertwining relations and the theory of Markov functions, which we will review. This talk is based on a joint work with Jonas Arista and Neil O’Connell: https://arxiv.org/abs/2203.14868.