This bulletin from the Florida Department of Revenue provides guidance on your situation, and also has more information specific to Virginia in the table. You may also find this Stack Exchange post helpful about a similar situation, in which someone else's lease payment went up after moving from Kentucky to Ohio.
Florida use tax of 6% generally applies to motor vehicles purchased in another state, territory of the U.S., or the District of Columbia, and subsequently titled, registered, or licensed in Florida.
Section 212.06(7), F.S., allows a credit to be given on motor vehicles brought into Florida where a like tax has been lawfully imposed and paid in another state, territory of the U.S., or the District of Columbia. Credit against Florida use tax and any discretionary sales surtax is given for a like tax paid in another state, whether the tax has been paid to that state, or to a county or city (local taxes) within that state. If the amount paid is equal to or greater than the amount imposed by Florida, no additional tax is due. If the amount is less than the amount imposed by Florida, only the difference between the two is due.
In addition, Section 212.06(8)(a), F.S., provides a presumption that motor vehicles used in another state, territory of the U.S., or the District of Columbia for six months or longer before being brought into Florida were not purchased for use in Florida. When documentary proof establishes such prior use no Florida tax is due.