Using scaling arguments and extensive numerical simulations, we study the 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. The 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 that 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 microscopic roughness.