  Three scalar effective field theories have special properties in terms of
non-linear symmetries, soft limits and on-shell constructability that arise
from their Goldstone nature: the non-linear $\sigma$-model, multi-DBI theory
and the special Galileon. We discuss how these theories are related via
flavour-kinematics duality, analogous to the colour-kinematics duality between
gravity and gauge theories. At the off-shell level, we identify a specific
mapping between the three theories that is crucially dependent on their
non-linear symmetries. Similarly, we demonstrate how the on-shell amplitudes
factorise into BCJ numerators describing flavour and a scalar version of
kinematics, naturally leading to the inclusion of graviton exchange in the
$SO(M,N)$ non-linear $\sigma$-model. Finally, we map those numerators onto each
other, and comment on a similar relation to tensor kinematics. Our results
highlight a common structure that underlies the physics of different Goldstone
modes.
