The reconfiguration graph of the k-colourings, denoted R k (G), is the graph whose vertices are the k-colourings of G and two colourings are adjacent in R k (G) if they differ in colour on exactly one vertex. In this paper, we investigate the connectivity and diameter of R k+1 (G) for a k-colourable graph G restricted by forbidden induced subgraphs. We show that R k+1 (G) is connected for every k-colourable H-free graph G if and only if H is an induced subgraph of P 4 or P 3 + P 1 . We also start an investigation into this problem for classes of graphs defined by two forbidden induced subgraphs. We show that if G is a k-colourable (2K 2 , C 4 )-free graph, then R k+1 (G) is connected with diameter at most 4n. Furthermore, we show that R k+1 (G) is connected for every k-colourable (P 5 , C 4 )-free graph G.