A rainbow blow-up lemma for almost optimally bounded edge-colourings

A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Koml\'os, S\'ark\"ozy and Szemer\'edi that applies to almost optimally bounded colourings. A corollary of this is that there exists a rainbow copy of any bounded-degree spanning subgraph $H$ in a quasirandom host graph $G$, assuming that the edge-colouring of $G$ fulfills a boundedness condition that is asymptotically best possible. This has many applications beyond rainbow colourings, for example to graph decompositions, orthogonal double covers and graph labellings.