  We study the null-controllability of some hypoelliptic quadratic parabolic
equations posed on the whole Euclidean space with moving control supports, and
provide necessary or sufficient geometric conditions on the moving control
supports to ensure null-controllability. The first class of equations is the
one associated to non-autonomous Ornstein-Uhlenbeck operators satisfying a
generalized Kalman rank condition. In particular, when the moving control
supports comply with the flow associated to the transport part of the
Ornstein-Uhlenbeck operators, a necessary and sufficient condition for
null-controllability on the moving control supports is established. The second
class of equations is the class of accretive non-selfadjoint quadratic
operators with zero singular spaces for which some sufficient geometric
conditions on the moving control supports are also given to ensure
null-controllability.
