I believe that the 14th amendment empowers and requires the federal government to protect RKBA from local infringement just as the feds protect freedom of religion, freedom of speech and of the press, and the rights to due process from local infringement.
Obviously, true respect for RKBA would no more require a permit or background checks than we tolerate permits or background checks to exercise other fundamental, enumerated rights.
Sadly, I do not believe the SCOTUS has yet defined/recognized RKBA in quite such broad terms. And so I expect that some kind of national reciprocity law is likely to be a necessary stepping stone along the path to full recognition of RKBA....unless we simply have no national protection until well after we get congress or the courts to recognize RKBA anywhere close to the terms we understand here. Background checks are a related, but separate issue in my mind.
It would be great if congress or the federal courts would step in and prevent law abiding citizens from facing criminal charges simply because they cross the border from Virginia to DC or Maryland with an otherwise legally carried firearm. Whether that should take the form of some kind of national reciprocity requirement, national constitutional carry requirements, national requirement to issue non-resident permits on a shall issue basis, or something else is probably more a matter of timing than anything else. Again, ultimately, we should have nationwide constitutional carry as the statutory law of the land to properly respect constitutional provisions.