We investigate the long time behavior of a passive particle evolving in a one-dimensional diffusive random environment, with diffusion constant $D$. We consider two cases: (a) The particle is pulled forward by a small external constant force, and (b) there is no systematic bias. Theoretical arguments and numerical simulations provide evidence that the particle is eventually trapped by the environment. This is diagnosed in two ways: The asymptotic speed of the particle scales quadratically with the external force as it goes to zero, and the fluctuations scale diffusively in the unbiased environment, up to possible logarithmic corrections in both cases. Moreover, in the large $D$ limit (homogenized regime), we find an important transient region giving rise to other, finite-size scalings, and we describe the cross-over to the true asymptotic behavior.