On preperiodic points for rational functions defined over $mathbb{F}_p(t)$

Let $Pin mathbb(P)_1(mathbb{Q})$ be a periodic point for a monic polynomial with coefficients in $mathbb{Z}$. With elementary techniques one sees that the minimal periodicity of $P$ is at most 2. Recently we proved a generalization of this fact to the set of all rational functions defined over $mathbb{Q}$ with good reduction everywhere (i.e. at any finite place of $mathbb{Q}$). The set of monic polynomials with coefficients in $mathbb{Z}$ can be characterized, up to conjugation by elements in PGL$_2(mathbb{Z}), as the set of all rational functions defined over $mathbb{Q}$ with a totally ramified fixed point in $mathbb{Q}$ and with good reduction everywhere. Let $p$ be a prime number and let $mathbb{F}_p$ be the field with $p$ elements. In the present paper we consider rational functions defined over the rational global function field $mathbb{F}_p(t)$ with good reduction at every finite place. We prove some bounds for the cardinality of orbits in $mathbb{F}_pcup{infty}$ for periodic and preperiodic points.