  We introduce an approach to compute the renormalisation group flow of
relational observables in quantum gravity which evolve from their microscopic
expressions towards the full quantum expectation value. This is achieved by
using the composite operator formalism of the functional renormalisation group.
These methods can be applied to a large class of relational observables within
a derivative expansion for different physical coordinate systems. As a first
application we consider four scalar fields coupled to gravity to represent the
physical coordinate frame from which relational observables can be constructed.
At leading order of the derivative expansion the observables are the inverse
relational metric and the relational scalar curvature. We evaluate their
scaling dimensions at the fixed point, both in the standard renormalisation
group scheme and in the essential scheme. This represents the first steps to
describe running observables within asymptotic safety; this treatment can be
generalised to other observables constructed from different tensors and in
different physical coordinate systems.
