Given an inclusion B subset of F of (graded) local nets, we analyse the structure of the corresponding inclusion of scaling limit nets B-0 subset of F-0, giving conditions, fulfilled in free field theory, under which the unicity of the scaling limit of F implies that of the scaling limit of B. As a byproduct, we compute explicitly the (unique) scaling limit of the fixpoint nets of scalar free field theories. In the particular case of an inclusion A subset of B of local nets with the same canonical field net F, we find sufficient conditions which entail the equality of the canonical field nets of A(0) and B-0.