Points of low degree on curves over function fields

We show that the geometric classification of smooth projective curves admitting infinitely many points of degree $d\leq 5$ extends from number fields to function fields of characteristic 0. Over number fields, this classification was established by Faltings for $d=1$, Harris--Silverman for $d=2$, Abramovich--Harris for $d=3,4$ and Kadets--Vogt for $d=4,5$. Our approach uses a specialization argument to reduce the problem over function fields to the number field case.