The full text of Feinstein's AWB bill
has been posted to her Senate website.
Note that the section on background checks for transfers of grandfathered assault weapons starts on page #119 of the PDF file, where is could easily be overlooked after the 93-page listing of exempt firearms.
The actual bill is not as vile as Feinstein has been claiming for the last month. Possession and transfers of weapons are allowed. I need to cross-reference the other laws that the background checks section cites, but it does not (after a very quick first read) appear to be NFA registration and taxation.
I have posted an abbreviated version
of the bill to Scribd that is only 6 pages long after omitting the lists of weapons and government exceptions.