We consider a model for the motion of an impurity interacting with two parallel, one-dimensional baths, described as two Tomonaga-Luttinger liquid systems. The impurity is able to move along the baths, and to jump from one to the other. We provide a perturbative expression for the evolution of the system when the impurity is injected in one of the baths, with a given wave packet. We obtain an approximation formally of infinite order in the impurity-bath coupling, which allows us to reproduce the orthogonality catastrophe. We monitor and discuss the dynamics of the impurity and its effect on the baths, in particular for a Gaussian wave packet. Besides the motion of the impurity, we also analyze the dynamics of the bath density and momentum density (i.e., the particle current), and show that it fits an intuitive semiclassical interpretation. We also quantify the correlation that is established between the baths by calculating the interbath, equal-time spatial correlation functions of both bath density and momentum, finding a complex pattern. We show that this pattern contains information on both the impurity motion and on the baths, and that these can be unveiled by taking appropriate "slices" of the time evolution.
