  We employ the curvature expansion of the quantum effective action for
gravity-matter systems to construct graviton-mediated scattering amplitudes for
non-minimally coupled scalar fields in a Minkowski background. By design, the
formalism parameterises all quantum corrections to these processes and is
manifestly gauge-invariant. The conditions resulting from UV-finiteness,
unitarity, and causality are analysed in detail and it is shown by explicit
construction that the quantum effective action provides sufficient room to meet
these structural requirements without introducing non-localities or higher-spin
degrees of freedom. Our framework provides a bottom-up approach to all quantum
gravity programs seeking for the quantisation of gravity within the framework
of quantum field theory. Its scope is illustrated by specific examples,
including effective field theory, Stelle gravity, infinite derivative gravity,
and Asymptotic Safety.
