  Pushable homomorphisms and the pushable chromatic number $\chi_p$ of oriented
graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably
observed that, for any oriented graph $\overrightarrow{G}$, we have
$\chi_p(\overrightarrow{G}) \leq \chi_o(\overrightarrow{G}) \leq 2
\chi_p(\overrightarrow{G})$, where $\chi_o(\overrightarrow{G})$ denotes the
oriented chromatic number of $\overrightarrow{G}$. This stands as first general
bounds on $\chi_p$. This parameter was further studied in later works.This work
is dedicated to the pushable chromatic number of oriented graphs fulfilling
particular degree conditions. For all $\Delta \geq 29$, we first prove that the
maximum value of the pushable chromatic number of an oriented graph with
maximum degree $\Delta$ lies between $2^{\frac{\Delta}{2}-1}$ and $(\Delta-3)
\cdot (\Delta-1) \cdot 2^{\Delta-1} + 2$ which implies an improved bound on the
oriented chromatic number of the same family of graphs. For subcubic oriented
graphs, that is, when $\Delta \leq 3$, we then prove that the maximum value of
the pushable chromatic number is~$6$ or~$7$. We also prove that the maximum
value of the pushable chromatic number of oriented graphs with maximum average
degree less than~$3$ lies between~$5$ and~$6$. The former upper bound of~$7$
also holds as an upper bound on the pushable chromatic number of planar
oriented graphs with girth at least~$6$.
