A growing number of biological, soft, and active matter systems are observed to exhibit anormal diffusive dynamics with a linear growth of the mean square displacement, yet with a non-Gaussian distribution of increments. Based on the Chubinsky-Slater idea of diffusing diffusivitiy, we here established and analyze a minimal model framework of diffusion processes with fluctuating diffusivity. In particular, we demonstrate the equivalence of diffusing diffusivity processes with a superstatistical approach with a distribution of diffusivities, at times shorther than the diffusivity correlation time. At longer times, a crossover to a Gaussian distributionwith an effective diffusivity emerges. Specifically, we established a subordination picture of Brownian yet not Gaussian diffusion processes, which can be used for a wide class of diffusivity fluctuation statistics. Our results are shown to be in excellent agreement with simulations and numerical evaluations.