  In this paper we show how to apply classical probabilistic tools for partial
sums $\sum_{j=0}^{n-1}\varphi\circ\tau^j$ generated by a skew product $\tau$,
built over a sufficiently well mixing base map and a random expanding dynamical
system. Under certain regularity assumptions on the observable $\varphi$, we
obtain a central limit theorem (CLT) with rates, a functional CLT, an almost
sure invariance principle (ASIP), a moderate deviations principle, several
exponential concentration inequalities and Rosenthal type moment estimates for
skew products with $\alpha, \phi$ or $\psi$ mixing base maps and expanding on
the average random fiber maps. All of the results are new even in the uniformly
expanding case. The main novelty here (contrary to \cite{ANV}) is that the
random maps are not independent, they do not preserve the same measure and the
observable $\varphi$ depends also on the base space. For stretched
exponentially $\al$-mixing base maps our proofs are based on multiple
correlation estimates, which make the classical method of cumulants applicable.
  For $\phi$ or $\psi$ mixing base maps, we obtain an ASIP and maximal and
concentration inequalities by establishing an $L^\infty$ convergence of the
iterates $\cK^n$ of a certain transfer operator $\cK$ with respect to a certain
sub-$\sig$-algebra, which yields an appropriate (reverse) martingale-coboundary
decomposition.
