  Motivated by the recent developments of gauge-covariant methods in the
phase-space, a systematic method is presented aiming at the generalisation of
the Moyal star-product to a non-Abelian gauge covariant one at any order. Such
an expansion contains some dressing of the bare particle model by the
gauge-fields explicitly, and might serve as a drastically simplifying tool for
the elaborations of gauge-covariant quantum transport models. In addition, it
might be of fundamental importance for the mathematical elaborations of gauge
theory using the strict or deformation quantisation principles. A few already
known examples of quantum kinetic theories are recovered without effort as an
illustration of the power of this tool. A gauge-covariant formulation taking
into account possible geometrical connections in both the position and momentum
spaces is also constructed at leading orders, with applications to the
generation of gauge-covariant effective theories in the phase-space. This paper
is devoted to the pedestrian elaboration of the gradient expansions. Their
numerous consequences will be explored in subsequent works.
