  Let $P$ be a parabolic subgroup of a connected simply connected complex
semisimple Lie group $G$. Given a compact K\"ahler manifold $X$, the
dimensional reduction of $G$-equivariant holomorphic vector bundles over
$X\times G/P$ was carried out by the first and third authors. This raises the
question of dimensional reduction of holomorphic principal bundles over
$X\times G/P$. The method used for equivariant vector bundles does not
generalize to principal bundles. In this paper, we adapt to equivariant
principal bundles the Tannakian approach of Nori, to describe the dimensional
reduction of $G$-equivariant principal bundles over $X\times G/P$, and to
establish a Hitchin--Kobayashi type correspondence. In order to be able to
apply the Tannakian theory, we need to assume that $X$ is a complex projective
manifold.
