Only if that union meets the new state's legal defintion of marriage. For same-sex marriages, that is not the case:
STATES THAT PREVENT RECOGNITION OF OUT-OF-STATE SAME-SEX MARRIAGE LICENSE
As you can see, they include both Florida and Illinois.
In the South, the standard reply to the argument that (fill in the black) was legal in Massachusetts is:
"Well, you're not in Massachusetts anymore, boy, and it ain't legal here!"