This paper studies compositions of independent random bundle maps \(F(x,a)=f_Fx,T_F(x)a\), \(x\in X\), \(a\in \mathbb R^d\), where \(X\) is a Borel subset of a Polish space, whose distributions form a stationary process. This specializes to the case of products of independent random matrices evolving by a stationary process and generalizes many results on products of random matrices. Ergodicity results are proved, the largest Lyapunov exponent is studied, and under certain conditions an asymptotically Gaussian distribution is obtained. This is applied to random harmonic functions and random continued fractions.