Products of $M$ i.i.d. non-Hermitian random matrices of size $N \times N$ relate Gaussian fluctuation of Lyapunov and stability exponents in dynamical systems (finite $N$ and large $M$) to local eigenvalue universality in random matrix theory (finite $M$ and large $N$). The remaining task is to study local eigenvalue statistics as $M$ and $N$ tend to infinity simultaneously, which lies at the heart of understanding two kinds of universal patterns. For products of i.i.d. complex Ginibre matrices, truncated unitary matrices and spherical ensembles, as $M+N\to \infty$ we prove that local statistics undergoes a transition when the relative ratio $M/N$ changes from $0$ to $\infty$: Ginibre statistics when $M/N \to 0$, normality when $M/N\to \infty$, and new critical phenomena when $M/N\to γ\in (0, \infty)$.