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A material system consisting of a lamellar grain structure adjacent to a large single grain is investigated. The system evolution is driven by changing of interface energy of the lamellar structure as well as by difference in bulk energies stored in the single grain and the lamellar grains. The triple junctions and the grain boundaries are assumed to have finite mobilities representing kinetic material parameters of system. A complete analysis of the kinetics of this system is provided, which involves several possible scenarios depending on the values of the geometrical and material parameters of the system. The scenarios are fully classified. Moreover, the analysis offers a way, how the values of the material parameters (interface energy densities, difference in bulk energies and mobilities) can be extracted from the measured system kinetics and geometry. © 2019 Acta Materialia Inc.