Even though the Constitution doesn't explicitly state that individuals have a right to travel, the Supreme Court has recognized the ability to move across state lines and localities as a fundamental, implied right.
One of the first cases to tackle the right to travel issue was Shapiro v. Thompson. The Connecticut Welfare Department denied Marie Thompson welfare benefits to her because she did not meet the requirement to live in the state for at least a year after moving from Dorchester, Massachusetts to Hartford, Connecticut. The state argued that the purpose of the law was to aid administrative efficiency and minimizing fraud. The court rejected Connecticut's argument as there was no compelling justification given by the state, and evidence did not fully support administrative ease nor minimize fraud in the program. Thus, the court struck down Connecticut's welfare law as it violated Thompson's Equal Protection Rights.
The Supreme Court didn't allow for an unlimited power to the right to travel because it upheld residency requirements for in-state college tuition.
Another major case to challenge a program based on the right to travel is Zobel v. Williams. The court relied on the implicit right to travel by striking down an Alaska law which gave out royalties for a state program based on how long an individual lived in the state. Instead of basing the decision on the Equal Protection Clause as the court had in Shapiro v. Thompson, the court based its decision on Article IV's Privileges and Immunities Clause.
A recent and significant case for the right to travel is seen in Saenz v. Roe.
California passed a law which paid welfare benefits to new residents in the same amount as the old state where the new individuals originated from. The court affirmed the right to travel as being an implicit right and stated that there were three components: the right to enter one state and leave another, the right to be treated as a welcome visitor rather than a hostile stranger and the right to be treated equally to native-born citizens.