The purpose of this work is to build a framework that allows for an in-depth study of various generalisations to inhomogeneous space of models of Borodin-Ferrari, Dieker-Warren, Nordenstam, Warren-Windridge of interacting particles in interlacing arrays, both in discrete and continuous time, involving both Bernoulli and geometric jumps. The models can in addition be either time-inhomogeneous or particle-inhomogeneous. We show that the correlation functions of these models are determinantal and using this we prove a short-time asymptotic for these dynamics to the discrete Bessel point process. We moreover prove a number of closely related results. We prove that the autonomous, inhomogeneous in space and time, TASEP-like and pushTASEP-like particle systems on the edges of the array have explicit transition kernels and that from any deterministic initial condition their distributions are marginals of a determinantal measure. We prove a novel duality relation between dynamics in inhomogeneous space and dynamics with inhomogeneities on the level of the array. We extend the work of Nordenstam on the shuffling algorithm for domino tilings of the Aztec diamond and its relation to push-block dynamics in interlacing arrays to general weights on the tilings and then connect this, for a special class of weights, back to our previous results. We also consider non-intersecting walks in inhomogeneous space and time with fixed starting and end points and obtain a formula for their correlation functions, involving among other ingredients, an explicit Riemann-Hilbert problem. We then prove a limit theorem for the bottom lines in this line-ensemble, under some technical conditions. The main computational tool throughout this work is a natural generalisation of a Toeplitz matrix, that we call inhomogeneous Toeplitz-like matrix $\mathsf{T}_{\mathbf{f}}$ with (a possibly matrix-valued) symbol $\mathbf{f}$.