Consider the discrete-time process formed by the singular values of products of random matrices, where time corresponds to the number of matrix factors. It is known due to Oseledets' theorem that under general assumptions, the Lyapunov exponents converge as the number of matrix factors tend to infinity. In this talk, we consider random matrices with distributional invariance under right multiplication by unitary matrices, which include Ginibre matrices and truncated unitary matrices. The corresponding singular value process is Markovian with additional structure that admits study via integrable probability techniques. In this talk, I will discuss recent results on the Lyapunov exponents in the setting where the number M matrix factors tend to infinity simultaneously with matrix sizes N. When this limit is tuned so that M and N grow on the same order, the limiting Lyapunov exponents can be described in terms of Dyson Brownian motion with a special drift vector, which in turn can be linked to a matrix-valued diffusion on the complex general linear group. We find that this description is universal, under general assumptions on the spectrum of the matrix factors.