Studying quantum field theories through geometric principles has revealed deep connections between physics and mathematics, including the discovery by Cachazo, Early, Guevara and Mizera (CEGM) of a generalization of biadjoint scalar amplitudes. However, extending this to generalizations of other quantum field theories remains a central challenge. Recently it has been discovered that the nonlinear sigma model (NLSM) emerges after a certain zero-preserving deformation from $\text{tr}(φ^3)$. In this work, we find a much richer story of zero-preserving deformations in the CEGM context, yielding generalized NLSM amplitudes. We prove an explicit formula for the residual embedding of an $n$-point NLSM amplitude in a mixed $n+2$ point generalized NLSM amplitude, which provides a strong consistency check on our generalization. We show that the dimension of the space of pure kinematic deformations is $\gcd(k,n)-1$, we introduce a deformation-compatible modification of the Global Schwinger Parameterization, and we include a new proof, using methods from matroidal blade arrangements, of the linear independence for the set of planar kinematic invariants for CEGM amplitudes. Our framework is compatible with string theory through recent generalizations of the Koba-Nielsen string integral to any positive configuration space $X^+(k,n)$, where the usual Koba-Nielsen string integral corresponds to $X(2,n) = \mathcal{M}_{0,n}$.