Besides the issue of defining a single "transformation map", there is an issue of defining a map which one can iterate.
Given a map $f : X \to Y$, one can iterate $f$ if and only if the condition $\text{range}(f) \subset X$ holds. You simply cannot iterate a map which fails to satisfy that condition.
I imagine that your intent is that any two of the Poincaré sections $P_1,...,P_n$ are disjoint. Therefore, if you fix $i \ne j \in \{1,...,n\}$ then $P_i$ and $P_j$ are disjoint and the map $f_{ij} : P_i \to P_j$ cannot be iterated.
But another thing one can do is to take the union $P = P_1 \cup ... \cup P_n$ to be the domain of a Poincare map $f : P \to P$. In that case it makes perfect sense to iterate the map $f$.
This iteration could be broken down as follows. Consider $x \in P$. We have $x \in P_i$ for a unique $i$. Flowing forward from $x$, we stop at the first moment we arrive at a point $y \in P$. We have $y \in P_j$ for a unique $j$. Therefore in this situation we have $f(x)=f_{ij}(x)=y$.
However, one might notice that $j$ depends on $x$. Choosing a different $x' \in P_i$ and flowing forward and stopping at the first moment we arrive at a point $y' \in P$, we may have $y' \in P_{j'}$ for $j' \ne j$.
What this shows us is that the iteratable map $f : P \to P$ can, with care, be patched together from pieces of the maps $f_{ij}$, but nonetheless it does not make sense to iterate the individual maps $f_{ij}$ when $i \ne j$.
