IF they do this, then your "Second Amendment rights" becomes a privilege...just like a drivers license. The Constitution does not give the right to own a horse or a car or any form of transportation or the right to operate said transportation...so it is a privilege. It DOES give the right to keep and bear arms. This enhanced background check is turning your "right" into a privilege controlled by the state. Which right will next become a privilege? Freedom of Speech?
I don't have a solution to the problem. I do know that prohibition and controls have never stopped criminal activity. In fact, it often enhances it and increases it based on the money that can be made trafficking any form of contraband.
Stopping a father or grandfather from transferring a firearm to a family member is completely against the Constitutional rights to own and bear arms and will absolutely not fix the problem...
Reading about the Court Cases and History of Discussion on the 2nd Amendment, it is easy to see this is NOT a simple issue.