Equivalence criteria are established for an effective Yukawa-type theory of composite fields representing two-particle fermion bound states with the original “microscopic” theory of interacting fermions based on the spectral decomposition of the 2-to-2 fermion scattering amplitude. Functional renormalization group equations of the effective theory are derived exploiting relations expressing the equivalence. The effect of truncating the spectral decomposition is investigated quantitatively on the example of the nonrelativistic bound states of two oppositely charged fermions.