Assault rifles are in homes - there's little that can be done about that. Ammunition is an item that is not so readily available on a whim of someone wanting to commit a mass shooting.
Restrictions must be put on the purchase of ammunition made for assault rifles, and on ammunition that can be loaded into a magazine or clip for any automatic weapons.
Require a waiting period - and verifiable identification to assure the ammunition is being purchased by the registered gun owner, and a 2nd background check to be done if the background check is older than 1 year.