  We prove the multisummability of the infinitesimal generator of unfoldings of
finite codimension tangent to the identity 1-dimensional local complex analytic
diffeomorphisms. We also prove the multisummability of Fatou coordinates and
extensions of the Ecalle-Voronin invariants associated to these unfoldings. The
quasi-analytic nature is related to the parameter variable. As an application
we prove an isolated zeros theorem for the analytic conjugacy problem.
  The proof is based on good asymptotics of Fatou coordinates and the
introduction of a new auxiliary tool, the so called multi-transversal flows.
They provide the estimates and the combinatorics of sectors typically
associated to summability. The methods are based on the study of the
infinitesimal stability properties of the unfoldings.
