You might be able to say something about Brauer-Manin obstructions for various moduli spaces of abelian varieties, especially modular curves. See math.mit.edu/~poonen/papers/heuristic.pdf

@DonuArapura theorem 1 of this paper seems relevant: arxiv.org/pdf/math/0605603

@JasonStarr I see my mistake— I intended the question to allow the base to be any smooth projective curve, which need not necessarily itself be smooth over $\mathbb{P}^1$. I apologize for the confusion and have edited the post

@AriyanJavanpeykar Every curve admits a nonconstant morphism to $\mathbb{P}^1$, so that one can always compose a map to a curve with such a morphism. I see now that this alone does not guarantee smoothness in positive characteristic as the map of curves may be inseparable. Is that what goes wrong in your example?

@willsawin thanks for the response. Could you please expand on how you conclude when the fibration splits after a finite cover?

I don’t think this answers the question, which is about abelian varieties over global fields, not over finite fields. Milne’s article does not discuss the Hasse-Weil L-function over global fields, and I do not think the fact that the higher cohomology groups are exterior powers of H^1 makes it straightforward to relate their global L-functions to that of H^1.

It would be helpful to state the full background in the question (what properties the map is supposed to satisfy), or at least say more specifically where this is located in the reference.

Aren’t algebraic groups which are isomorphic over a finite extension already isomorphic over $k$? Since they both have $k$-rational points they are trivial torsors. Perhaps you only intend for $G’$ to become an algebraic group upon base change to $K$?

I assume you meant for $f$ to be meromorphic. What you want is probably here: en.wikipedia.org/wiki/….