  The standard theoretical framework to deal with weak decays of heavy mesons
is the so-called weak effective Hamiltonian. It involves the short-distance
Wilson coefficients, which depend on the renormalisation scale $\mu$. For
specific calculations one has to evolve the Wilson coefficients down from the
electroweak scale $\mu=M_W$ to the typical mass scale of the decay under
consideration. This is done by solving a renormalisation group equation for the
effective operator basis. In this paper the results of a consistent two-step
running of the $c \to u \,\ell^+\ell^-$ Wilson coefficients for dimension-6
operators are presented. This running involves the intermediate scale $\mu=m_b$
(with $M_W > m_b > m_c$) where the bottom quark is integrated out. The matching
coefficients and anomalous dimensions are taken to the required order by
generalizing and extending results from $b \to s$ or $s \to d$ transitions
available in the literature.
