  We consider vector-valued solutions to a linear transmission problem, and we
prove that Lipschitz-regularity on one phase is transmitted to the next phase.
More exactly, given a solution $u:B_1\subset \mathbb{R}^n \to \mathbb{R}^m$ to
the elliptic system \begin{equation*} \mbox{div} ((A + (B-A)\chi_D )\nabla u) =
0 \quad \text{in }B_1, \end{equation*} where $A$ and $B$ are Dini continuous,
uniformly elliptic matrices, we prove that if $\nabla u \in L^{\infty} (D)$
then $u$ is Lipschitz in $B_{1/2}$. A similar result is also derived for the
parabolic counterpart of this problem.
