  We study the Renormalisation Group (RG) running of the quark mass, for
$N_f=3$ QCD with Wilson fermions in a mixed action setup, with standard
Schr\"odinger Functional (SF) boundary conditions for sea quarks and chirally
rotated Schr\"odinger Functional ($\chi$SF) boundary conditions for valence
quarks. This necessitates the tuning of the boundary factor $z_f(g_0^2)$ of the
$\chi$SF valence action, in order to ensure that QCD symmetries are fully
recovered in the continuum. The properties of this novel setup are monitored
through the ratios $Z_S/Z_P$ and $\Sigma_S/\Sigma_P$ of the renormalisation
parameters and step scaling functions of the scalar and pseudoscalar densities.
Where comparison is possible, our $Z_S/Z_P$ results are found to agree with
previous determinations, based on a mass ratio method arXiv:1906.03445 and Ward
identities arXiv:2005.01352, arXiv:2101.10969, with Schr\"odinger Functional
boundary conditions. The behaviour of $\Sigma_S/\Sigma_P$ confirms the
theoretical expectations of $\chi$SF QCD, related to the restoration of the
theory's symmetries in the continuum limit. From the step scaling function of
the pseudoscalar density we obtain the quark mass RG-running function from
hadronic to perturbative energy scales. This is fully compatible with the
earlier result obtained in a similar setup for Wilson quarks with Schr\"odinger
Functional boundary conditions arXiv:1802.05243 and provides a strong
universality test for the two lattice setups.
