  In this paper we consider a family of Dirac-type operators on fibration $P
\to B$ equivariant with respect to an action of an etale groupoid. Such a
family defines an element in the bivariant $K$ theory. We compute the action of
the bivariant Chern character of this element on the image of Connes' map
$\Phi$ in the cyclic cohomology. A particular case of this result is Connes'
index theorem for etale groupoids in the case of fibrations.
