  We study the motion of an impurity in a two-leg ladder interacting with two
fermionic baths along each leg, a simple model bridging cold atom quantum
simulators with an idealised description of the basic transport processes in a
layered heterostructure. Using the linked-cluster expansion we obtain exact
analytical results for the single-particle Green's function and find that the
long-time behaviour is dominated by an intrinsic orthogonality catastrophe
associated to the motion of the impurity in each one-dimensional chain. We
explore both the case of two identical legs as well as the case where the legs
are characterised by different interaction strengths: in the latter case we
observe a subleading correction which can be relevant for intermediate-time
transport at an interface between different materials. In all the cases we do
not find significant differences between the intra- and inter-leg Green's
functions in the long-time limit.
