  The asymptotic safety scenario in gravity is accessed within the systematic
vertex expansion scheme for functional renormalisation group flows put forward
in \cite{Christiansen:2012rx,Christiansen:2014raa}, and implemented in
\cite{Christiansen:2015rva} for propagators and three-point functions. In the
present work this expansion scheme is extended to the dynamical graviton
four-point function. For the first time, this provides us with a closed flow
equation for the graviton propagator: all vertices and propagators involved are
computed from their own flows.
  In terms of a covariant operator expansion the current approximation gives
access to $\Lambda$, $R$, $R^2$ as well as $R_{\mu\nu}^2$ and higher derivative
operators. We find a UV fixed point with three attractive and two repulsive
directions, thus confirming previous studies on the relevance of the first
three operators. In the infrared we find trajectories that correspond to
classical general relativity and further show non-classical behaviour in some
fluctuation couplings.
  We also find signatures for the apparent convergence of the systematic vertex
expansion. This opens a promising path towards establishing asymptotically safe
gravity in terms of apparent convergence.
