  Let $K$ be a number field and $S$ a finite set of places of $K$ that contains
all of the archimedean places. Let $\varphi: \mathbb{P}^1 \to \mathbb{P}^1$ be
a rational map of degree $d \geq 2$ defined over $K$. Given $\alpha \in
\mathbb{P}^1(K)$ non-preperiodic and $\beta \in \mathbb{P}^1(K)$
non-exceptional, we prove an upper bound of the form $O(|S|^{1+\epsilon})$ on
the number of points in the forward orbit of $\alpha$ that are $S$-integral
relative to $\beta$, extending results of Hsia--Silverman [HS11]. When
$\varphi$ is a power or Latt\`es map, we obtain a stronger upper bound of the
form $O(|S|)$. We also prove uniform bounds when $\varphi$ is a polynomial,
extending results of Krieger et al [KLS+15].
