The main focus of this work is the analysis of the asymptotic behavior of deterministic logit dynamics in heterogeneous routing games defined on two-terminal directed multigraphs. We provide three distinct results. The first one states that this dynamics admits a globally asymptotically stable fixed point when noise is sufficiently high. The second result, instead, shows that the fixed points of the dynamics always approach the set of Wardrop equilibria of the associated routing game, as noise vanishes. Finally, the third and last result says that fixed points are locally asymptotically stable, again for a sufficiently low noise level, when they converge to Wardrop equilibria consisting in flow distributions in which every users' population has a single optimal path to travel along..