Since nobody else has mentioned this, let me point out the article of Bogomolov and Tschinkel.

MR2154371 (2006e:14024) Reviewed
Bogomolov, Fedor(1-NY-X); Tschinkel, Yuri(D-GTN)
Rational curves and points on K3 surfaces. (English summary)
Amer. J. Math. 127 (2005), no. 4, 825–835.
14G05 (11G35 14J28)
http://arxiv.org/pdf/math/0310254.pdf

Let $Z$ be a Kummer K3 surface with a specified very ample divisor class $D$, let $Y$ be $\mathbb{P}^1$ with one marked point $0$, and let $X$ be $Y\times Z$ with its obvious projection $\text{pr}_Y$ to $Y$, with the $Y$-ample divisor class $\text{pr}_Z^*D$, and with an arbitrary $K$-rational point $x_0$ in the fiber over $0$ marked. Then Bogomolov and Tschinkel prove that for every finite field reduction $R/\mathfrak{m}$, there exists a section $s_{\mathfrak{m}}$ sending $0$ to $x_0$ and having positive degree with respect to $\text{pr}_Z^* D$, i.e., the section is not "horizontal". They point out that it may well be that the same holds over $\overline{K}$, but nobody really knows this, and some people are skeptical. So this is an example worth bearing in mind.