Reaction-diffusion processes occur in a wide variety of physical and biological settings. When the saturation of the diffusion flux, as is to be expected of a physical process at high gradients, is incorporated into these processes, it may cause a fundamental change in the ensuing morphology, since now not only discontinuous equilibria become admissible, but they may emerge instead of travelling waves (TWs). A class of such processes, modelled by the equation ut ≤ [Q(ux)]x - β2f(u), is studied both analytically and numerically. The diffusion flux Q is assumed to be a bounded increasing function with a sufficiently fast saturation rate, β > 0 is a constant, and a typical f(u) ≡ f(u; α1, ..., αN) with f(0; •) ≤ f(1; •) ≤ f(αi; •) ≤ 0, i ≤ 1, ..., N has a sequence of control parameters αi ∈ (0, 1). We show that the conventional equilibria kinks connecting upstream with downstream states may, when the parameter β exceeds a critical threshold, have a discontinuous part. Moreover, while in the conventional case of a linear Q when the total reaction is nonzero TWs form, a bounded Q enables discontinuous equilibria, provided β is sufficiently large. Remarkably, when the number of possible constant equilibria is greater than two, the pattern may admit more than one discontinuity. Uniqueness of such states depends on the class of initial data assumed. We also numerically demonstrate that these new equilibria are robust and are (strong) attractors for a wide class of initial data.