The FFL should check just before releasing the firearm, but it is not required. The DOJ will tell you that you have to have access to the DROS system in order to release a firearm, but I don't know where that is actually required.
I have heard of some places getting dropped off the system due to permits expiring and was recently told that 70 places just dropped off. In some cases it is not the fault of the place since the city might have issued the new permit in time, in other cases it can be because the place had not paid all the fees.
A rejection can come at any time and I have seen that. The rejection can be for many reasons, not just a background check, but things such as DMV issues. From what I have heard, the DOJ will call before putting it on the DROS system, which is a bit silly since the DOJ might not get the FFL on the phone as the FFL could be working, but that seems to be the way it is done.
FFL Transfer/Special Order Dealer since 1993.
Net-FFL list maintainer.
Never try to teach a pig to sing. You waste your time and you annoy the pig. - Robert A. Heinlein