Distances between distributions via Stein's method

We build on the formalism developed in Ernst et al. (First order covariance inequalities via Stein’s method, 2019) to propose new representations of solutions to Stein equations. We provide new uniform and nonuniform bounds on these solutions (a.k.a. Stein factors). We use these representations to obtain representations for differences between expectations in terms of solutions to the Stein equations. We apply these to compute abstract Stein-type bounds on Kolmogorov, total variation and Wasserstein distances between arbitrary distributions. We apply our results to several illustrative examples and compare our results with current literature on the same topic, whenever possible. In all occurrences our results are competitive.