A numerical monoid is an additive submonoid of the non-negative integers. Given a numerical monoid S, consider the family of "shifted" monoids M n obtained by adding n to each generator of S. In this paper, we examine minimal relations among the generators of M n when n is sufficiently large, culminating in a description that is periodic in the shift parameter n. We explore several applications to computation, combinatorial commutative algebra, and factorization theory.