We investigate the emergence of non-linear diffusivity in kinetically constrained, one-dimensional symmetric exclusion processes satisfying the gradient condition. Recent developments introduced new gradient dynamics based on the Bernstein polynomial basis, enabling richer diffusive behaviours but requiring adaptations of existing techniques. In this work, we exploit these models to generalise the Porous Media Model to non-integer parameters and establish simple conditions on general kinetic constraints under which the empirical measure of a perturbed version of the process converges. This provides a robust framework for modelling non-linear diffusion from kinetically constrained systems.