In this paper, we present a thorough analysis of the future asymptotic dynamics of spatially homogeneous cosmological models of Bianchi type VI₀. Each of these models converges to a flat Kasner solution (Taub solution) for late times; we give detailed asymptotic expansions describing this convergence. In particular, we prove that the future asymptotics of Bianchi type VI₀ solutions cannot be approximated in any way by Bianchi type II solutions, which is in contrast to Bianchi type VIII and IX models (in the direction toward the singularity). The paper contains an extensive introduction where we put the results into a broader context. The core of these considerations consists in the fact that there exist regions in the phase space of Bianchi type VIII models where solutions can be approximated, to a high degree of accuracy, by type VI₀ solutions. The behavior of solutions in these regions is essential for the question of 'locality', i.e., whether particle horizons form or not. Since Bianchi type VIII models are conjectured to be important role models for generic cosmological singularities, our understanding of Bianchi type VI₀ dynamics might thus be crucial to help to shed some light on the important question of whether to expect generic singularities to be local or not.