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 transversal or 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.