We derive parity-even graviton bispectra in the Effective Field Theory of
Inflation (EFToI) to all orders in derivatives. Working in perturbation theory,
we construct all cubic interactions that can contribute to tree-level graviton
bispectra, showing that they all come from EFToI operators containing two or
three powers of the extrinsic curvature and its covariant derivatives: all
other operators can be removed by field redefinitions or start at higher-order
in perturbations. For operators cubic in the extrinsic curvature, where the
single-clock consistency relations are satisfied without a correction to the
graviton two-point function, we use the Manifestly Local Test (MLT) to
efficiently extract the effects of evolving graviton fluctuations to the end of
inflation. Despite the somewhat complicated nature of the bulk interactions,
the final boundary correlators take a very compact form. For operators
quadratic in the extrinsic curvature, the leading order bispectra are a sum of
contact and single exchange diagrams, which are tied together by spatial
diffeomorphisms, and to all orders in derivatives we derive these bispectra by
computing the necessary bulk time integrals. For single exchange diagrams we
exploit factorisation properties of the bulk-bulk propagator for massless
gravitons and write the result as a finite sum over residues. Perhaps
surprisingly, we show these single exchange contributions have only
total-energy poles and also satisfy the MLT