We study the diffusive dynamics of a polymer driven through a narrow channel with which it has strong interactions. We show that when there is a chemical potential drop across the channel, there is a range of polymer lengths in which the system is approximately one-dimensional and translationally invariant. In this regime, a coarse-grained description is used to study the distribution of times for the polymer to pass through the channel. We also introduce a more microscopic model, and use it to estimate the parameter values in the coarse-grained description. Finally, we present a comparison with recent experiments.