In this paper, we develop a theory to study the imposed axial salt-concentration-gradient-driven ionic diffusioosmosis (IDO) in soft nanochannels or nanochannels grafted with end-charged polyelectrolyte (PE) brushes. Our analysis first quantifies the diffusioosmotically induced electric field, which is primarily dictated by the imposed concentration gradient (CG) with little contribution of the induced osmosis. This induced electric field triggers an electroosmotic (EOS) transport, while the net diffusioosmotic (DOS) transport results from a combination of this EOS transport and a chemiosmotic (COS) transport arising from the pressure gradient induced by the applied CG. Our results demonstrate that the DOS transport is massively enhanced in nanochannels grafted with PE brushes with weak grafting density stemming from the significantly enhanced EOS transport caused by the localization of the EOS body force away from the nanochannel walls. This augmentation is even stronger for cases where the COS transport aids the EOS transport. On the other hand, the DOS transport gets severely reduced in nanochannels grafted with dense PE brushes owing to the severity of the brush-induced additional drag force. We anticipate that these findings will help to unravel an entirely new understanding of induced electrokinetic transport in soft nanochannels.