The function field analog of the theorem of Mazur-Kamienny-Merel (giving a universal bound in $[K:\mathbb{Q}]$ for the size of the $K$-rational torsion of an elliptic curve) is immediate just from the fact that the modular varieties involved happen to be algebraic curves with gonality going to infinity with the level. In fact the bound here turns out to be polynomial in the gonality of the function field, something which is conjectured to hold but not known in the original setting.

Has there been any progress on the higher dimensional version of the problem in function fields? The simplest statement here is the uniform boundedness of the group of $k(T)$-rational torsion of a traceless abelian surface over $k(T)$.