  Given graphs $G_1,\ldots,G_s$ all on a common vertex set and a graph $H$ with
$e(H) = s$, a copy of $H$ is \emph{transversal} or \emph{rainbow} if it
contains one edge from each $G_i$. We establish a stability result for
transversal Hamilton cycles: the minimum degree required to guarantee a
transversal Hamilton cycle can be lowered as long as the graph collection
$G_1,\ldots,G_n$ is far in edit distance from several extremal cases. We obtain
an analogous result for Hamilton paths. The proof is a combination of our newly
developed regularity-blow-up method for transversals, along with the absorption
method.
