  Given two $k$-dicolourings of a digraph $D$, we prove that it is
PSPACE-complete to decide whether we can transform one into the other by
recolouring one vertex at each step while maintaining a dicolouring at any step
even for $k=2$ and for digraphs with maximum degree $5$ or oriented planar
graphs with maximum degree $6$. A digraph is said to be $k$-mixing if there
exists a transformation between any pair of $k$-colourings. We show that every
digraph $D$ is $k$-mixing for all $k\geq \delta^*_{\min}(D)+2$, generalizing a
result due to Dyer et al. We also prove that every oriented graph $\vec{G}$ is
$k$-mixing for all $k\geq \delta^*_{\max}(\vec{G}) +1$ and for all $k\geq
\delta^*_{\rm avg}(\vec{G})+1$. We conjecture that, for every digraph $D$, the
dicolouring graph of $D$ on $k\geq \delta_{\min}^*(D)+2$ colours has diameter
at most $O(|V(D)|^2)$ and give some evidences. We first prove that the
dicolouring graph of any digraph $D$ on $k\geq 2\delta_{\min}^*(D) + 2$ colours
has linear diameter, extending a result from Bousquet and Perarnau. We also
prove that the conjecture is true when $k\geq
\frac{3}{2}(\delta_{\min}^*(D)+1)$. Restricted to the special case of oriented
graphs, we prove that the dicolouring graph of any subcubic oriented graph on
$k\geq 2$ colours is connected and has diameter at most $2n$. We conjecture
that every non $2$-mixing oriented graph has maximum average degree at least
$4$, and we provide some support for this conjecture by proving it on the
special case of $2$-freezable oriented graphs. More generally, we show that
every $k$-freezable oriented graph on $n$ vertices must contain at least $kn +
k(k-2)$ arcs, and we give a family of $k$-freezable oriented graphs that reach
this bound. In the general case, we prove as a partial result that every non
$2$-mixing oriented graph has maximum average degree at least $\frac{7}{2}$.
