  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
$\mathbf{f}$.
