I hesitate to take too seriously or opine too firmly on the rumored plans to deploy members of the National Guard from some (red) states to other (blue) states around the country — partly inspired by recent deployments in DC. But if the federal government envisions deploying the National Guard from some states (under state control) to others, I wonder about two potential constitutional issues.
One is whether those deployments violate Article IV of the Constitution. Article IV commits to the United States to "protect each [State] against invasion.” (And for good measure Article I also forbids each State to “engage in War, unless actually invaded, or in such imminent Danger as will not admit of delay.”). Does the deployment of one state’s troops into another state over the latter state’s objection constitute an invasion?
A second question is whether federal support for those deployments are within the Article I militia power. Article I allows Congress “To provide for calling forth the Militia to execute the Laws of the Union, suppress Insurrections and repel Invasions” and Article II makes the President Commander-in-Chief if but only if its called forth. But to provide federal support for the interstate deployment of the militia — without calling it into federal control — sits at least somewhat uneasily with this Clause.
(These two questions might also be related. The more one tries to avoid the Article I problem by emphasizing that the states are acting on their own sovereignty here, the more the Article IV problems are heightened. Just as one state traditionally did not have to respect the judgments of another state if those judgments were rendered in violation of state sovereignty, one might ask — if a state deploys troops to another state against that state’s will, how is the latter state entitled to respond?)
Both of these problems are eased if the National Guard is called forth into federal service, but my understanding is that doing so implicates other federal laws such as the Posse Commitatus Act, and was not done in DC, so I thought it was worth flagging these constitutional questions. I certainly hope they do not arise!
For decades now, there has been in force an Interstate Compact wherein at least 49 (last I checked, and it may now be 50) States agreed to assist each other in natural disasters and emergencies with their respective National Guards to reinforce and augment each other. This was definitely the case in 2005 during responses to Hurricane Katrina. The Interstate Compact was duly authorized/ratified by the Congress pursuant to Article I, section 10 of the Constitution.
One suspects this will also suffice to legalize the contribution by red States of their National Guards to assist the federal government, certainly in DC if not the other states as well.
The National Guard is a part of the state militia, but it's also more than that. For historical reasons recounted in Perpich v. Department of Defense, the National Guard also became part of the armed forces of the United States so the President could take the Guard where the militia couldn't go.
It's not just when in active duty. Members of the National Guard take two oaths: a State oath and a federal oath. Normally, they wear both hats.
Critical to your question, however, is that when ordered to active federal duty, a member is (usually) "relieved from duty in the National Guard of his State." 32 USC § 325(a).
So if members of the National Guard of Texas are ordered to active federal duty in Illinois, they don't go as members of the militia, but as members of the US armed forces.
FYI: Not my expertise either, but someone mentioned this to me years ago, so I knew to look for it.
References
https://tile.loc.gov/storage-services/service/ll/usrep/usrep496/usrep496334/usrep496334.pdf (Perpich, discussing the history)
https://constitution.congress.gov/browse/essay/artI-S8-C16-1/ALDE_00013673/#ALDF_00027673 (more history)
https://www.law.cornell.edu/uscode/text/32/101 (definitions)
https://www.law.cornell.edu/uscode/text/10/246 (definition of militia)
https://www.law.cornell.edu/uscode/text/10/12406 (call into federal service)
https://www.law.cornell.edu/uscode/text/32/304 (enlistment oath)