  Earth's rotation about the Sun produces an annual modulation in the expected
scattering rate at direct dark matter detection experiments. The annual
modulation as a function of the recoil energy $E_\text{R}$ imparted by the dark
matter particle to a target nucleus is expected to vary depending on the
detector material. However, for most interactions a change of variables from
$E_\text{R}$ to $v_\text{min}$, the minimum speed a dark matter particle must
have to impart a fixed $E_\text{R}$ to a target nucleus, produces an annual
modulation independent of the target element. We recently showed that if the
dark matter-nucleus cross section contains a non-factorizable target and dark
matter velocity dependence, the annual modulation as a function of
$v_\text{min}$ can be target dependent. Here we examine more extensively the
necessary conditions for target-dependent modulation, its observability in
present-day experiments, and the extent to which putative signals could
identify a dark matter-nucleus differential cross section with a
non-factorizable dependence on the dark matter velocity.
