gaseousclay - excluding some specific enhancement of the background check you mention and a government data-base that is essentially what happens now.
If the police choose to run a trace on a firearm the gun manufacturer is contacted and they tell what wholesaler they sold it to, then the wholesaler is contacted and they say what retailer (local FFL or LGS) the FFL then checks his form 4473's and tells them who he sold it to - name and address, make and model and serial number of gun to verify. There are many local jurisdictions and I believe some states that prohibit transfer of firearms to unlicensed let alone prohibited individuals and it is a crime to straw purchase - but I would be surprised if there are more than a handful of prosecutions based on such laws in the whole country in a years time - straw purchasers just say the gun was stolen. Usually it seems they discover it was stolen right after being informed it was used in a crime.
If the police choose to run a trace on a firearm the gun manufacturer is contacted and they tell what wholesaler they sold it to, then the wholesaler is contacted and they say what retailer (local FFL or LGS) the FFL then checks his form 4473's and tells them who he sold it to - name and address, make and model and serial number of gun to verify. There are many local jurisdictions and I believe some states that prohibit transfer of firearms to unlicensed let alone prohibited individuals and it is a crime to straw purchase - but I would be surprised if there are more than a handful of prosecutions based on such laws in the whole country in a years time - straw purchasers just say the gun was stolen. Usually it seems they discover it was stolen right after being informed it was used in a crime.