Depends on the TA.
The one I am using right now, a very large one, is always offering assorted OBC's, free DSC's, gift cards. They won't adjust those because your fare dropped - they will substitute whatever their current promotions are for what you originally had.
I booked a cruise last July. Got some deals from the TA and OBC from NCL.
A few weeks later, TA offered the same credits (combined about $500) plus the DSC (worth $600 on mine). So, I rebooked, but NCL had dropped their OBC ($200). So I saved a net $400.