We study global fluctuations for singular values of $M$-fold products of several right-unitarily invariant $N \times N$ random matrix ensembles. As $N \to \infty$, we show the fluctuations of their height functions converge to an explicit Gaussian field, which is log-correlated for $M$ fixed and has a white noise component for $M \to \infty$ jointly with $N$. Our technique centers on the study of the multivariate Bessel generating functions of these spectral measures, for which we prove a central limit theorem for global fluctuations via certain conditions on the generating functions. We apply our approach to a number of ensembles, including square roots of Wishart, Jacobi, and unitarily invariant positive definite matrices with fixed spectrum, using a detailed asymptotic analysis of multivariate Bessel functions to verify the necessary conditions.