The Lüroth semigroups of a curve over a non-algebraically closed field

Let C⊂P2 be a smooth curve defined over a non-algebraically closed field K. We study the Lüroth semigroups of C over K, i.e. the set L′(C,K) of all degrees of finite morphisms C→P1 defined over K and the set L(C,K) of all degrees >0 of some spanned line bundle on C defined over K. If K is infinite, then L′(C,K) = L(C,K), but for every prime power q ≠ 2 there is a smooth plane curve C defined over Fq with L′(C,Fq)⊆L(C,Fq) and C(Fq)≠∅. If C is a smooth plane curve, then L(C,K) determines (in several ways) if C(K) ≠ ∅.