We investigate whether the boundaries of a region of plasma in the ionosphere of different density than its surroundings will drift relative to the background ambipolar drift and, if so, how the drift depends on the degree of density enhancement, or on the altitude. There are analytical solutions for discrete circular features both with and without neutral collisions. We find that the drift is proportional to the density difference, which suggests that where density gradients occur they should tend to steepen on one side of a patch while they are weakened on the other. This may have relevance to the morphology of polar ionospheric patches and auroral arcs. The drift of the boundaries of a patch is seen to be distinct from the ion particle drift; nevertheless it appears that density structures can retain a cohesive shape even while the plasma that constitutes the enhancement or depletion moves into, through, and out of the structure again. There is also an altitude dependence (through the ion-neutral collision frequency) which may generate E-region shears and concomitant field-aligned currents above the edges of sharp density gradients.