  Let $D=(V,A)$ be a digraph. We define $\Delta_{\max}(D)$ as the maximum of
$\{ \max(d^+(v),d^-(v)) \mid v \in V \}$ and $\Delta_{\min}(D)$ as the maximum
of $\{ \min(d^+(v),d^-(v)) \mid v \in V \}$. It is known that the dichromatic
number of $D$ is at most $\Delta_{\min}(D) + 1$. In this work, we prove that
every digraph $D$ which has dichromatic number exactly $\Delta_{\min}(D) + 1$
must contain the directed join of $\overleftrightarrow{K_r}$ and
$\overleftrightarrow{K_s}$ for some $r,s$ such that $r+s = \Delta_{\min}(D) +
1$, except if $\Delta_{\min}(D) = 2$ in which case $D$ must contain a digon. In
particular, every oriented graph $\vec{G}$ with $\Delta_{\min}(\vec{G}) \geq 2$
has dichromatic number at most $\Delta_{\min}(\vec{G})$. Let $\vec{G}$ be an
oriented graph of order $n$ such that $\Delta_{\min}(\vec{G}) \leq 1$. Given
two 2-dicolourings of $\vec{G}$, we show that we can transform one into the
other in at most $n$ steps, by recolouring one vertex at each step while
maintaining a dicolouring at any step. Furthermore, we prove that, for every
oriented graph $\vec{G}$ on $n$ vertices, the distance between two
$k$-dicolourings is at most $2\Delta_{\min}(\vec{G})n$ when $k\geq
\Delta_{\min}(\vec{G}) + 1$. We then extend a theorem of Feghali, Johnson and
Paulusma to digraphs. We prove that, for every digraph $D$ with
$\Delta_{\max}(D) = \Delta \geq 3$ and every $k\geq \Delta +1$, the
$k$-dicolouring graph of $D$ consists of isolated vertices and at most one
further component that has diameter at most $c_{\Delta}n^2$, where $c_{\Delta}
= O(\Delta^2)$ is a constant depending only on $\Delta$.
