  We show the smoothness over the affine line of the Hodge moduli space of
logarithmic t-connections of coprime rank and degree on a smooth projective
curve with geometrically integral fibers over an arbitrary Noetherian base.
When the base is a field, we also prove that the Hodge moduli space is
geometrically integral. Along the way, we prove the same results for the
corresponding moduli spaces of logarithmic Higgs bundles and of logarithmic
connections.
  We use smoothness to derive specialization isomorphisms on the etale
cohomology rings of these moduli spaces; this includes the special case when
the base is of mixed characteristic. In the special case where the base is a
separably closed field of positive characteristic, we show that these
isomorphisms are filtered isomorphisms for the perverse filtrations associated
with the corresponding Hitchin-type morphisms.
