I recommend that you look at my joint paper with Tom Graber.

MR3114946 Pending
Graber, Tom(1-CAIT); Starr, Jason Michael(1-SUNYS)
Restriction of sections for families of abelian varieties. (English summary) A celebration of algebraic geometry, 311–327,
Clay Math. Proc., 18, Amer. Math. Soc., Providence, RI, 2013.
14K12 (14C05)

In particular, surjectivity frequently fails. For instance, starting with $X$ the projective linear system of plane cubics containing one specified point, and denoting by $\mathcal{A}$ the universal plane cubic over $X$ (with its specified point as the "origin"), let $S$ be a general linear $2$-plane inside $X$, and let $C$ be a general line inside $S$. Then $C$ gives a pencil of plane cubics that has 9 base points (one of which is our specified point). These base points give lots of rational points in $\mathcal{A}(C)$, and these need not lift (of course your base field is "small", so this might take some work to completely justify).

What Tom and I do show is that, if you use "line pairs" rather than "lines", and if you allow $C$ to be a "sufficiently general" line pair, then the restriction map is a bijection.