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Brownian Motion, with some persistence in the direction of motion, typically known as active Brownian Motion, has been observed in many significant chemical and biological transport processes. Here, we present a model of drifted Brownian Motion that considers a nonlinear stochastic drift with constant or fluctuating diffusivity. The interplay between nonlinearity and structural heterogeneity of the environment can explain three essential features of active transport. These features, which are commonly observed in experiments and molecular dynamics simulations, include transient superdiffusion, ephemeral non-Gaussian displacement distribution, and non-monotonic evolution of non-Gaussian parameter. Our results compare qualitatively well with experiments of self-propelled particles in simple hydrogen peroxide solutions and molecular dynamics simulations of self-propelled particles in more complex settings such as viscoelastic polymeric media.