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