The position $x(t)$ of a particle diffusing in a one-dimensional uncorrelated and time dependent random medium is simply Gaussian distributed in the typical direction, i.e. along the ray $x=v_0 t$, where $v_0$ is the average drift. However, it has been found that it exhibits at large time sample to sample fluctuations characteristic of the KPZ universality class when observed in an atypical direction, i.e. along the ray $x = v t$ with $v \neq v_0$. Here we show, from exact solutions, that in the moderate deviation regime $x - v_0 t \propto t^{3/4}$ these fluctuations are precisely described by the finite time KPZ equation, which thus describes the crossover between the Gaussian typical regime and the KPZ fixed point regime for the large deviations. This confirms heuristic arguments given in [2]. These exact results include the discrete model known as the Beta RWRE, and a continuum diffusion. They predict the behavior of the maximum of a large number of independent walkers, which should be easier to observe (e.g. in experiments) in this moderate deviations regime.