In the U.S. there is no centralized agency that oversees accreditation of higher education institutions and the power and responsibility of accreditation are decentralized and distributed to various regional agencies. I know there are agencies on the national level but I think they are mostly specialized. So if an accredited college/university (liberal arts or offering similarly general education/degrees) moves from one region to a different region does it need to start from scratch with the process and get re-accredited by the agency in charge of the new region?
I am not sure of all the details, but it appears that since a controversial regulatory change that went into effect in 2020, "regional accreditors" are no longer legally restricted to operate within a fixed geographical region. See 84 FR 58834. See also an analysis from Inside Higher Ed (maybe paywalled). In effect, as I understand it, each agency is now allowed to declare for itself what region it wishes to cover, and they are not forbidden from overlapping.
It seems this change effectively erased the legal distinction between regional and national accreditors, lumping them together in a new category of "institutional accreditors" though I think people still know which is which.
As a result:
It is now possible for a university to switch between "regional" accreditors without moving. It appears, for instance, that the University of Arizona is currently in the process of doing just that, leaving HLC for WSCUC. Formerly WSCUC only covered California, Hawaii, and the US territories in the Pacific, but in 2020 it unilaterally dropped its geographical restrictions.
So procedures evidently do exist for an institution to switch accreditors, and your hypothetical Interstate U could presumably follow them.
On the other hand, it would also be possible for Interstate U's current accreditor to expand their territory to cover the new campus location, in which case I assume Interstate U could move without switching at all.
I don't know how this would have been handled prior to 2020. I'm not aware of it ever having come up.