  We use the functional renormalization group equation for quantum gravity to
construct a non-perturbative flow equation for modified gravity theories of the
form $S = \int d^dx \sqrt{g} f(R)$. Based on this equation we show that certain
gravitational interactions monomials can be consistently decoupled from the
renormalization group (RG) flow and reproduce recent results on the asymptotic
safety conjecture. The non-perturbative RG flow of non-local extensions of the
Einstein-Hilbert truncation including $\int d^dx \sqrt{g} \ln(R)$ and $\int
d^dx \sqrt{g} R^{-n}$ interactions is investigated in detail. The inclusion of
such interactions resolves the infrared singularities plaguing the RG
trajectories with positive cosmological constant in previous truncations. In
particular, in some $R^{-n}$-truncations all physical trajectories emanate from
a Non-Gaussian (UV) fixed point and are well-defined on all RG scales. The RG
flow of the $\ln(R)$-truncation contains an infrared attractor which drives a
positive cosmological constant to zero dynamically.
