Making rules for naturalization is an act of becoming a citizen *after* immigration. The constitution does not empower congress any authority over the migration of people, only the states can control immigration within their borders.
Also the constitution only gives congress the power to make rules, not carry them out. Acts of naturalzation was always carried out by the states who often applied their own rules above what congress enacted.
It's been a while since states set rules for naturalization.
http://www.usconstitution.com/NaturalizationActof1795.htm
THE NATURALIZATION ACT OF 1795
An Act to establish an uniform Rule of Naturalization; and to repeal the Act heretofore passed on that Subject. For carrying into complete effect the power given by the constitution, to establish an uniform rule of naturalization throughout the United States;
I know of no state that has any mechanism in place to keep track of who moves in or out, and what their status is.
And a very good thing, too.