I will talk about a general class of integrable models of interacting particles in inhomogeneous space, containing various types of inhomogeneous pushTASEPs and zero range processes, and how they are connected to determinantal point processes, random walks conditioned to never intersect and random tilings of the Aztec diamond with inhomogeneous weights. The integrability of these models comes from a natural generalisation of Toeplitz matrices which satisfy certain intertwining relations.