Let $f: X \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ compact and convex.
From Brouwer fixed-point theorem, $f$ admits a fixed point ($\bar{x} \in X$ such that $f(\bar{x}) = \bar{x}$).
Further assume that $f$ is piecewise-affine, i.e. $f(x) = A_i x + b_i$ if $x \in X_i \subseteq X$, where $X = \bigcup_{i} X_i$, with $X_i \cap X_j = \varnothing$ $\forall i \neq j$.
Are there fixed-point iterations for such specific $f$s generating a sequence $\{x_k\}_k$ converging to a fixed point?
Not really an answer but a rather long comment. I think it is enough to consider $f_i$ (that is $f_i$ restrict to $X_i$). In this case you get the fixed points are the vectors that solve $(I-A_i)^{-1}b_i$. If $I-A_i$ doesn't have an inverse, the use the pseudo-inverse. Also $f_i$ may not have a fixed point or as you pointed $(I-A_i)^{-1}b_i\not\in X_i$.
In terms of iterations: $f_i^1 x= A_i x +b_i$, $f_i^2 x= A_i (A_i x + b_i)+b_i=A_i^2x+A_ib_i+b_i$ and so on so: $$f_i^n x = A_i^n x + \sum_{k=0}^{n-1} A_i^k b_i.$$ If $||A_i||<1$ then $||A_i^n||\rightarrow 0$ as $N\rightarrow\infty$ and also in this case $\sum_{k=0}^{\infty}A_i^k=(I-A)^{-1}$.
To check if $||A_i||$ it is sufficient (and necessary) all eigenvalues of $A_i$ be such that $|\lambda_i|<1$.