  Over the last years the Asymptotic Safety program has matured into a serious
candidate for a quantum theory of gravity compatible with observations. The
rapid technical progress in computing renormalisation group flows for gravity
and gravity-matter systems in the non-perturbative regime has put many
interesting physical questions within reach. In particular, the construction of
the non-perturbative quantum corrections to the propagation of fields on a
fluctuating spacetime allows addressing the effective propagation of matter on
a quantum spacetime or the possible resolution of spacetime singularities based
on first principle computations. In this article, we assemble a technical
toolbox for carrying out investigations on this promising research frontier. As
a specific example we present results for the momentum-dependent two-point
function for a scalar field induced by the quantum fluctuations of the
underlying geometry in a self-consistent way.
