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

Large deviations, moment estimates and almost sure invariance principles for skew products with mixing base maps and expanding on the average fibers