This paper addresses the deterministic translational-rotational motion of a freely-draining rigid dumbbell that is freely suspended in pressure-driven Stokes flow through a periodically corrugated pore or channel. As is demonstrated by numerical integration of the coupled, nonlinear equations of motion within a sinusoidally corrugated, axisymmetric pore, passage of the dumbbell through successive converging and diverging local flow environments engenders rotary fluctuations relative to the local streamlines that gradually increase in amplitude, over many pore wavelengths, until it eventually flips over. Then the dumbbell slowly realigns itself with the flow, with diminishing fluctuations-whereupon the whole cycle begins again.To analyze the cumulative effect of zero-mean rotary fluctuations, we first consider a simpler problem in which a prolate body of zero axis ratio is subjected to externally applied, harmonic rotary oscillations while immersed in steady, homogeneous shear. Two-timescale asymptotics combined with inner-outer matching yield a leading-order formula for the orientational history that agrees very closely with the numerical solution for this case. In particular, the non-oscillatory (slow) component of the behavior is asymptotically equivalent to a Jeffery orbit described by a nonzero effective axis ratio.Subsequently the hydrodynamic mechanism responsible for moving the dumbbell (of dimensionless length 2l) out of alignment with the local streamlines is analyzed for a two-dimensional, periodically corrugated channel, with accounting for (i) the leading effect of spatial inhomogeneity of the flow field, and (ii) the first two corrections in (small) angle of misalignment. For sufficiently short times following the initial condition of perfect alignment with the flow, the trajectory of the dumbbell＇s midpoint is effectively decoupled from the orientational history. These approximations lead to a Riccati equation, for which the coefficients are periodic functions under a subtle assumption involving the flow field. A two-scale analysis then explains the numerically observed ord (l(-1)) scaling of the distance traveled between successive flips. Motivation for the required assumption, and approximations of the hydrodynamic coefficients, are obtained from lubrication theory. When a sinusoidally corrugated, two-dimensional channel is chosen to match the meridian-plane generator of the axisymmetric pore, the asymptotic orientational history for the former and the numerical solution for the latter are found to appear remarkably similar.Our analysis is placed in context with regard to Lagrangian unsteadiness and the general dynamical-systems theory advanced in a series of papers by A. J. Szeri, S. Wiggins, W. J. Milliken and/or L. G. Leal [J. Fluid Mech., 228, 207-241 (1991); 237, 33-56 (1992); 250, 143-167 (1993)] for microstructure in "complex" flows. The orientational dynamics considered here have important implications for deterministic cross-stream migration of elastic dumbbells [Nitsche, L. C., AIChE J., 42, 613-622 (1996)].