Multiplying sequence of $3\times 3$ matrices

Question

Suppose I have a column stochastic matrix $$A(x) = \begin{pmatrix}1-a_{1}(x) & a_{2}(x) & 0 \\ a_{1}(x) & 1- a_{2}(x)-a_3(x) & a_{4}(x) \\ 0 & a_{3}(x) & 1-a_{4}(x) \end{pmatrix}.$$ Is there some nice formula that I can use to find each entry of the matrix $A(x)A(f(x))A(f(f(x)))\cdot \cdots \cdot A(f^n(x))$ where $f:\mathbb{R}\to \mathbb{R}$? In the $2\times 2$ case I have been able to do this but was wondering if it were possible to generalise this.