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-average random fiber maps. All of the results are new even in the uniformly expanding case. The main novelty here (in contrast to [2]) 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
${\alpha }$-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
${\mathcal K}^{\,n}$ of a certain transfer operator
${\mathcal K}$ with respect to a certain sub-
${\sigma }$-algebra, which yields an appropriate (reverse) martingale-coboundary decomposition.