  We discuss the renormalisation properties of the full set of $\Delta F=2$
operators involved in BSM processes, including the definition of RGI versions
of operators that exhibit mixing under RG transformations. As a first step for
a fully non-perturbative determination of the scale-dependent renormalization
factors and their runnings, we introduce a family of appropriate Schr\"odinger
Functional schemes, and study them in perturbation theory. This allows, in
particular, to determine the NLO anomalous dimensions of all $\Delta F=1,2$
operators in these schemes. Finally, we discuss the systematic uncertainties
related to the use of NLO perturbation theory for the RG running of four-quark
operators to scales in the GeV range, in both our SF schemes and standard
$\overline{MS}$ and RI-MOM schemes. Large truncation effects are found for some
of the operators considered.
