A local asymptotic theory of adaptive processes of growth with general increments is developed for the case when a terminal set consists of more than one connected component. The notions of an attainable and unattainable component are introduced. Sufficient conditions for attainability and unattainability are derived. The limit theorems are applied in the investigation of the rate of convergence to singleton stable components. The relation between the obtained results and the study of asymptotic properties of stochastic quasi-gradient algorithms in non-convex multiextremum problems is discussed. Specifically, the developed approach is used to explore the limit behavior of iterations in the Fabian modification of the Kiefer-Wolfowitz algorithm
