  We prove that the approximate null-controllability with uniform cost of the
hypoelliptic Ornstein-Uhlenbeck equations posed on $\mathbb R^n$ is
characterized by an integral thickness geometric condition on the control
supports. We also provide associated quantitative weak observability estimates.
This result for the hypoelliptic Ornstein-Uhlenbeck equations is deduced from
the same study for a large class of non-autonomous elliptic equations from
moving control supports. We generalize in particular results known for
parabolic equations posed on $\mathbb R^n$, for which the approximate
null-controllability with uniform cost is ensured by the notion of thickness,
which is stronger that the integral thickness condition considered in the
present work. Examples of those parabolic equations are the fractional heat
equations associated with the operator $(-\Delta)^s$, in the regime $s\geq1/2$.
Our strategy also allows to characterize the approximate null-controllability
with uniform cost from moving control supports for this class of fractional
heat equations.
