Nonlocality theorem concerns writing a covariance as $C(a,b)=\int A(a,x)A(b,x)dx$ with A in {1,-1}.
This is not without remembering the weak formulation of FEM with a continuous basis index a or b.
But as quantum mechanics shows this is not always possible and one shall write $A(a,b,x)$ according to mainstream.
How would this be implemented in finite element method basis functions ? Does this mean somehow that the basis functions "know" with which other function it will be multiplied and integrated ?