  Left-Right (LR) models are extensions of the Standard Model where left-right
symmetry is restored at high energies, and which are strongly constrained by
kaon mixing described in the framework of the $|\Delta S|=2$ effective
Hamiltonian. We consider the short-distance QCD corrections to this Hamiltonian
both in the Standard Model (SM) and in LR models. The leading logarithms
occurring in these short-distance corrections can be resummed within a
rigourous Effective Field Theory (EFT) approach integrating out heavy degrees
of freedom progressively, or using an approximate simpler method of regions
identifying the ranges of loop momentum generating large logarithms in the
relevant two-loop diagrams. We compare the two approaches in the SM at
next-to-leading order, finding a very good agreement when one scale dominates
the problem, but only a fair agreement in the presence of a large logarithm at
leading order. We compute the short-distance QCD corrections for LR models at
next-to-leading order using the method of regions, and we compare the results
with the EFT approach for the $WW'$ box with two charm quarks (together with
additional diagrams forming a gauge-invariant combination), where a large
logarithm occurs already at leading order. We conclude by providing
next-to-leading-order estimates for $cc$, $ct$ and $tt$ boxes in LR models.
