  We investigate the convergence in distribution of sequential empirical
processes of dependent data indexed by a class of functions F. Our technique is
suitable for processes that satisfy a multiple mixing condition on a space of
functions which differs from the class F. This situation occurs in the case of
data arising from dynamical systems or Markov chains, for which the
Perron--Frobenius or Markov operator, respectively, has a spectral gap on a
restricted space. We provide applications to iterative Lipschitz models that
contract on average.
