Over at Cato, George Mason Law professor  Ilya Somin takes a look at the Constitution and concludes it's not incompatible with open borders:

 

Does the Constitution Give the Federal Government Power Over Immigration? Cato Unbound, September 12, 2018

 

I would credit Congress' power over "naturalization" in Article I, Section 8 with broader reach than he does. 

 

But Congress or the President's power over who can immigrate seems so basic to sovereignty that maybe that's why the Framers weren't more explicit.