  We compute the one- and two-loop RG flow of integrable $\sigma$-models with
Poisson-Lie symmetry. They are characterised by a twist function with $2N$
simple poles/zeros and a double pole at infinity. Hence, they capture many of
the known integrable deformations in a unified framework, which has a geometric
interpretation in terms of surface defects in a 4D Chern-Simons theory. We find
that these models are one-loop renormalisable and present a very simple
expression for the flow of the twist function. At two loops only models with
$N$=1 are renormalisable. Applied to the $\lambda$-deformation on a semisimple
group manifold, our results reproduce the $\beta$-functions in the literature.
