  The only known fully ghost-free and consistent Lorentz-invariant kinetic term
for a graviton (or indeed for any spin-2 field) is the Einstein-Hilbert term.
Here we propose and investigate a new family of candidate kinetic interactions
and their extensions to derivative interactions involving several spin-2
fields. These new terms generically break diffeomorphism invariance(s) and as a
result can lead to the propagation of 5 degrees of freedom for a single spin-2
field - analogous to ghost-free Massive Gravity. We discuss under what
circumstances these new terms can be used to build healthy effective field
theories and in the process establish the `Jordan' and `Einstein' frame
pictures for Massive-, Bi- and Multi-Gravity.
