Recent theoretical modeling offers a unified picture for the description of stochastic processes characterized by a crossover from anomalous to normal behavior. This is particularly welcome, as a growing number of experiments suggest the crossover to be a common feature shared by many systems: in some cases the anomalous part of the dynamics amounts to a Brownian yet non-Gaussian diffusion; more generally, both the diffusion exponent and the distribution may deviate from normal behavior in the initial part of the process. Since proposed theories work at a mesoscopic scale invoking the subordination of diffusivities, it is of primary importance to bridge these representations with a more fundamental, ``microscopic'' description. We argue that the dynamical behavior of macromolecules during simple polymerization processes provide suitable setups in which analytic, numerical, and particle-tracking experiments can be contrasted at such a scope. Specifically, we demonstrate that Brownian yet non-Gaussian diffusion of the center of mass of a polymer is a direct consequence of the polymerization process. Through the kurtosis, we characterize the early-stage non-Gaussian behavior within a phase diagram, and we also put forward an estimation for the crossover time to ordinary Brownian motion.