This is a tremendously important question, as Ilya's and my posts noted in May. Justice Kavanaugh's concurrence in Dobbs today expressly noted:
[A]s I see it, some of the other abortion-related legal questions raised by today's decision are not especially difficult as a constitutional matter. For example, may a State bar a resident of that State from traveling to another State to obtain an abortion? In my view, the answer is no based on the constitutional right to interstate travel.
I expect this is fairly important because I assume that Chief Justice Roberts and the three dissenters (Justices Breyer, Sotomayor, and Kagan) and Justice Jackson, who will replace Justice Breyer, would likely go along with Kavanaugh on this. (Indeed, some of the other Justices in the majority might, too.) It's not certain, of course, especially as to Roberts; still, Kavanaugh's pronouncement here strikes me as highly significant.