  We revisit the so-called Cat-and-Mouse Markov chain, studied earlier by
Litvak and Robert (2012). This is a 2-dimensional Markov chain on the lattice
$\mathbb{Z}^2$, where the first component (the cat) is a simple random walk and
the second component (the mouse) changes when the components meet. We obtain
new results for two generalisations of the model. Firstly, in the 2-dimensional
case we consider far more general jump distributions for the components and
obtain a scaling limit for the second component. When we let the first
component be a simple random walk again, we further generalise the jump
distribution of the second component. Secondly, we consider chains of three and
more dimensions, where we investigate structural properties of the model and
find a limiting law for the last component.
