  In this paper we study a sequence of Bouchaud trap models on $\mathbb{Z}$
with drift. We analyze the possible scaling limits for a sequence of walks,
where we make the drift decay to 0 as we rescale the walks. Depending on the
speed of the decay of the drift we obtain three different scaling limits. If
the drift decays slowly as we rescale the walks we obtain the inverse of an
\alpha$-stable subordinator as scaling limit. If the drift decays quickly as we
rescale the walks, we obtain the F.I.N. diffusion as scaling limit. There is a
critical speed of decay separating these two main regimes, where a new process
appears as scaling limit. This critical speed is related to the index $\alpha$
of the inhomogeneity of the environment.
