Could travel be a way to express oneself?
It appears that the "leading" source of Freedom of Movement is the Privileges and Immunities Clause (Art. IV, S.2 , Cl. 1) of the US Constitution, that "The Citizens of each State shall be entitled to all Privileges and Immunities of Citizens in the several States". See Crandall v. State of Nevada, 73 US 35: "We are all citizens of the United States, and as members of the same community must have the right to pass and repass through every part of it without interruption, as freely as in our own states" (quoted from an earlier case); US v. Wheeler, 254 U.S. 281
In all the states, from the beginning down to the establishment of the Articles of Confederation, the citizens possessed the right, inherent in citizens of all free governments, peacefully to dwell within the limits of their respective states, to move at will from place to place therein, and to have free ingress thereto and egress therefrom
citing the Privileges & Immunities clause as the constitutional foundation.
However, this article, sect. IB notes a number of additional constiutional sources:
Various Justices at various times have suggested no fewer than seven different sources: the Article IV Privileges and Immunities Clause, the Fourteenth Amendment Privileges and [sic] Immunities Clause, a conception of national citizenship said to be implicit in the structural logic of the Constitution itself, the Commerce Clause, the Equal Protection Clause, and each of the Due Process Clauses.
Edwards v. California, 314 U.S. 160 relates freedom of movement to the Commerce Clause, Aptheker v. Sec’y of State, 378 U.S. 500 points us to the Due Process clause. The argument hasn't apparently been made based on the First Amendment, since there are better arguments.