No it doesn't.
If someone is in state A and has criminal conviction, his name goes into the III (Interstate Identification Index) which is operated by the FBIs NCIC. If the person goes to state B to purchase a firearm, NICS (or the respective state POC) will run his information through the various person files at NCIC, including the III. If they get a hit on the III, they will delay the NICS and pull the information from the state system in state A to see whether it represents a prohibiting condition and then either approves or denies the transfer. If they deny the transfer, the name, social etc. go into the 'NICS denial file' at NCIC and is available for any further inquiries.
The system is far from perfect. One problem is that it doesn't run an IBIS check to weed out people involved in cross-border trafficking and people who shouldn't be here. Also, the mental health information varies state by state, so even though someone like Seung-Hui Cho had been found to be mentally defective, his information was not recorded in the database due to the way the reporting law was written at the time.