  Nonlinear sigma model (NLSM) based on the coset $\text{SU}(N)\times
\text{SU}(N)/\text{SU}(N)$ exhibits several intriguing features at the leading
${\cal O}(p^2)$ in the derivative expansion, such as the flavor-kinematics
duality and an extended theory controlling the single and triple soft limits.
In both cases the cubic biadjoint scalar theory plays a prominent role. We
extend these features in two directions. First we uncover a new extended theory
for $\text{SO}(N+1)/\text{SO}(N)$ NLSM at ${\cal O}(p^2)$, which is a cubic
bifundamental/biadjoint scalar theory. Next we provide evidence for
flavor-kinematics dualities up to ${\cal O}(p^4)$ for both $\text{SU}(N)$ and
$\text{SO}(N)$ NLSM's. In particular, we introduce a new duality building block
based on the symmetric tensor $\delta^{ab}$ and demonstrate several
flavor-kinematics dualities for 4-point amplitudes, which precisely match the
soft blocks employed to soft-bootstrap the NLSM's up to ${\cal O} (p^4)$.
