The transport of suspended Brownian particles dc-driven along corrugated narrow channels is numerically investigated in the regime of finite damping. We show that inertial corrections cannot be neglected as long as the width of the channel bottlenecks is smaller than an appropriate particle diffusion length, which depends on the the channel corrugation and the drive intensity. Being such a diffusion length inversely proportional to the damping constant, transport through sufficiently narrow obstructions turns out to be always sensitive to the viscosity of the suspension fluid. The inertia corrections to the transport quantifiers, mobility and diffusivity, markedly differ for smoothly and sharply corrugated channels.
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