  We study Miyaoka-type semistability criteria for principal Higgs G-bundles E
on complex projective manifolds of any dimension. We prove that E has the
property of being semistable after pullback to any projective curve if and only
if certain line bundles, obtained from some characters of the parabolic
subgroups of G, are numerically effective. One also proves that these
conditions are met for semistable principal Higgs bundles whose adjoint bundle
has vanishing second Chern class. In a second part of the paper, we introduce
notions of numerical effectiveness and numerical flatness for principal (Higgs)
bundles, discussing their main properties. For (non-Higgs) principal bundles,
we show that a numerically flat principal bundle admits a reduction to a Levi
factor which has a flat Hermitian-Yang-Mills connection, and, as a consequence,
that the cohomology ring of a numerically flat principal bundle with real
coefficients is trivial. To our knowledge this notion of numerical
effectiveness is new even in the case of (non-Higgs) principal bundles.
