  We study an open question at the interplay between the classical and the
dynamical Mordell-Lang conjectures in positive characteristic. Let $K$ be an
algebraically closed field of positive characteristic, let $G$ be a finitely
generated subgroup of the multiplicative group of $K$, and let $X$ be a
(irreducible) quasiprojective variety defined over $K$. We consider $K$-valued
sequences of the form $a_n:=f(\varphi^n(x_0))$, where $\varphi\colon
X\rightarrow X$ and $f\colon X\rightarrow\mathbb{P}^1$ are rational maps
defined over $K$ and $x_0\in X$ is a point whose forward orbit avoids the
indeterminacy loci of $\varphi$ and $f$. We show that the set of $n$ for which
$a_n\in G$ is a finite union of arithmetic progressions along with a set of
upper Banach density zero. In addition, we show that if $a_n\in G$ for every
$n$ and the $\varphi$ orbit of $x$ is Zariski dense in $X$ then {there is} a
multiplicative torus $\mathbb{G}_m^d$ and maps $\Psi:\mathbb{G}_m^d \to
\mathbb{G}_m^d$ and $g:\mathbb{G}_m^d \to \mathbb{G}_m$ such that $a_n = g\circ
\Psi^n(y)$ for some $y\in \mathbb{G}_m^d$. We then describe various
applications of our results.
