  Given graphs $G_1,\ldots,G_s$ all on the same vertex set and a graph $H$ with
$e(H) \leq s$, a copy of $H$ is transversal or rainbow if it contains at most
one edge from each $G_c$. When $s=e(H)$, such a copy contains exactly one edge
from each $G_i$. We study the case when $H$ is spanning and explore how the
regularity blow-up method, that has been so successful in the uncoloured
setting, can be used to find transversals. We provide the analogues of the
tools required to apply this method in the transversal setting. Our main result
is a blow-up lemma for transversals that applies to separable bounded degree
graphs $H$.
  Our proofs use weak regularity in the $3$-uniform hypergraph whose edges are
those $xyc$ where $xy$ is an edge in the graph $G_c$. We apply our lemma to
give a large class of spanning $3$-uniform linear hypergraphs $H$ such that any
sufficiently large uniformly dense $n$-vertex $3$-uniform hypergraph with
minimum vertex degree $\Omega(n^2)$ contains $H$ as a subhypergraph. This
extends work of Lenz, Mubayi and Mycroft.
