We study a general limiting framework for the convergence of sequences of additive functionals of diffusions to Lévy subordinators, and provide explicit sufficient conditions that both ensure convergence and characterize the law of the limit. As an application, we identify a novel limiting regime for Wright-Fisher and Feller diffusions in the reflecting case and describe the corresponding limiting subordinator. This work is motivated by, and has applications in, neuroscience, where reflected diffusions are used to parametrize synchrony in doubly-stochastic models of spiking activity.