  Using scaling arguments and extensive numerical simulations, we study
dynamics of a tracer particle in a corrugated channel represented by a periodic
sequence of broad chambers and narrow funnel-like bottlenecks enclosed by a
hard-wall boundary. A tracer particle is affected by an external force pointing
along the channel, and performs an unbiased diffusion in the perpendicular
direction. We present a detailed analysis a) of the distribution function of
the height above the funnel's boundary at which the first crossing of a given
bottleneck takes place, and b) of the distribution function of the first
passage time to such an event. Our analysis reveals several new features of the
dynamical behaviour which are overlooked in the studies based on the
Fick-Jacobs approach. In particular, trajectories passing through a funnel
concentrate predominantly on its boundary, which makes first-crossing events
very sensitive to the presence of binding sites and a microscopic roughness.
