Consider the composition $F = f_{\omega_n} \circ \dots \circ f_{\omega_1}$ of $n$ random i.i.d. area-preserving surface maps, and iterate this composition. Such "semirandom" dynamics can be thought of as a toy model for iterating Poincaré maps describing repeated passage through multiple resonances. If $\{f_\omega\}$ expands on average, i.i.d. iterations are exponentially mixing, as recently shown by deWitt and Dolgopyat. In particular, $F$ satisfies a mixing estimate exponential in $n$. We show that a similar mixing estimate also applies to the iterates $F^m$ of $F$, up to an exponentially large threshold on $m$. Namely, there is $\alpha > 0$ such that for all large enough $n$ with the probability $ > 1 - e^{-\alpha n}$ the map $F$ satisfies the following estimate simultaneously for all integers $m \in [1, e^{\alpha n}]$: \[ \left| \int \phi \cdot (\psi \circ F^m) \; d \vol - \int \phi \; d \vol \int \psi \; d \vol \right| \le e^{-\alpha n}||\phi||_{C^\beta}||\psi||_{C^\beta}, \] where $\phi, \psi \in C^\beta$ are any observables for fixed $0 < \beta < 1$, and $\vol$ denotes the normalized area. This implies a Law of Large Numbers and a Central Limit Theorem for the Birkhoff sums $\phi + \phi \circ F + \dots + \phi \circ F^{M_n}$ when $n \to \infty$ and $M_n \to \infty$, $M_n < e^{\delta n}$, where $\delta>0$ is small enough. Joint work in progress with D. Dolgopyat, B. Fayad, and J. Paradela.