No, there is a right to travel freely from state to state at will and nowhere is it written in the constitution that one must leave fundamental civil rights behind.
The definition and extend of those "fundamental civil rights" is what is being 'explored' in light of the recent 2nd amendment cases from SCOTUS.
o form criteria has been established for what level of intrusion is allowed (strict scrutiny, compelling interest, etc.).
While there are may years of jurisprudence for the 1st and 4th amendments, there is barely any on the 2nd.