The kinetics of "nucleation-and-branching" rate-limited solid-state reactions and phase transformations (which include processes that are governed by nucleation and nuclei multiplication but not nuclei growth or crystal growth) are described using coupled rate equations. We treat such processes as occurring in two steps. The first step, nucleation, is assumed to be first order in the reagent. The second step, involving the multiplication of product nuclei (or "nuclei branching"), is considered to be autocatalytic; i.e., the rate of this step depends on the relative amounts of both the reagent and the product. The general rate equation developed on the basis of this two-step model allows greater flexibility in the fitting of sigmoidal kinetic curves than does the widely used Prout-Tompkins equation. Similarities between our model and the recently described general Prout-Tompkins (GPT) equation are discussed.