  We revisit and extend the `Effective field theory for massive gravitons'
constructed by Arkani-Hamed, Georgi and Schwartz in the light of recent
progress in constructing ghost-free theories with multiple interacting spin-2
fields. We show that there exist several dual ways of restoring gauge
invariance in such multi-gravity theories, find a generalised Fierz-Pauli
tuning condition relevant in this context and highlight subtleties in demixing
tensor and scalar modes. The generic multi-gravity feature of scalar mixing and
its consequences for higher order interactions are discussed. In particular we
show how the decoupling limit is qualitatively changed in theories of
interacting spin-2 fields. We relate this to dRGT (de Rham, Gabadadze, Tolley)
massive gravity, Hassan-Rosen bigravity and the multi-gravity constructions by
Hinterbichler and Rosen. As an additional application we show that EBI
(Eddington-Born-Infeld) bigravity and higher order generalisations thereof
possess ghost-like instabilities.
