  We have calculated continuum limit step scaling functions of bilinear and
four-fermion operators renormalized in a Rome-Southampton scheme using various
smearing prescriptions for the gauge field. Also, for the first time, we have
calculated non-perturbative anomalous dimensions of operators renormalized in a
Rome-Southampton scheme. The effect of such smearing first enters connected
fermionic correlation functions via radiative corrections. We use off-shell
renormalisation as a probe, and observe that the upper edge of the
Rome-Southampton window is reduced by link smearing. This can be interpreted as
arising due to the fermions decoupling from the high momentum gluons and we
observe that the running of operators with the scale at large lattice momenta
shows enhanced lattice artefacts. We find that the effect is greater for HEX
smearing than for Stout smearing, but that in both cases additional care must
be taken when using off-shell renormalisation with smeared gauge fields
compared to thin link simulations.
