We use numerical simulations to examine the effects of radial migration on the vertical structure of galaxy disks. The simulations follow three exponential disks of different mass but similar circular velocity, radial scalelength, and (constant) scale height. The disks develop different non-axisymmetric patterns, ranging from feeble, long-lived multiple arms to strong, rapidly-evolving few-armed spirals. These fluctuations induce radial migration through secular changes in the angular momentum of disk particles, mixing the disk radially and blurring pre-existing gradients. Migration affects primarily stars with small vertical excursions, regardless of spiral pattern. This "provenance bias" largely determines the vertical structure of migrating stars: inward migrators thin down as they move in, whereas outward migrators do not thicken up but rather preserve the disk scale height at destination. Migrators of equal birth radius thus develop a strong scale-height gradient, not by flaring out as commonly assumed, but by thinning down as they spread inward. Similar gradients have been observed for low-[$\alpha$/Fe] mono-abundance populations (MAPs) in the Galaxy but our results argue against interpreting them as a consequence of radial migration. This is because outward migration does not lead to thickening, implying that the maximum scale height of any population should reflect its value at birth. In contrast, Galactic MAPs have scale heights that increase monotonically outwards, reaching values that greatly exceed those at their presumed birth radii. Given the strong vertical bias affecting migration, a proper assessment of the importance of radial migration in the Galaxy should take carefully into account the strong radial dependence of the scale heights of the various stellar populations.