Let 𝔽 be a finite field and C,D smooth, geometrically irreducible, proper curves over 𝔽 and set K=𝔽(D). We consider Brauer–Manin and abelian descent obstructions to the existence of rational points and to weak approximation for the curve C⊗ 𝔽 K. In particular, we show that Brauer–Manin is the only obstruction to weak approximation and the Hasse principle in the case that the genus of D is less than that of C. We also show that we can identify the points corresponding to non-constant maps D→C using Frobenius descents.