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Phase coarsening in multiphase materials is a fundamental and crucial process in the study of microstructure evolution, for which a thorough understanding of kinetics can make a significant contribution to material processing and performance design. In this work, a new framework in the dimensionless Lifshitz-Slyozov-Wagner space is developed to study the kinetics of transient coarsening co-controlled by interface and matrix diffusion. The dynamic equations for individual particles are derived from the thermodynamic extremal principle, in consideration of energy dissipations by matrix diffusion, interface migration/reaction, and trans-interface diffusion. The effects of initial particle size distribution and finite volume fractions are clarified. Different from the conventional viewpoints, the time for transient coarsening is found to change non-monotonically with the width and tail length of the initial distribution. The ultralong transient coarsening with 'quasi-steady' distributions can occur for systems initiated from both the 'wide & long tail' and 'narrow & short tail' distributions. Furthermore, this work proves the existence of a unique attractor state for the steady stage and reveals the numerical origin of 'quasi-steady' distributions, which answers Brown's unsolved puzzle for years [LC Brown. Acta Metall 1989;37:71]. Finally, an increase in volume fraction is shown to directly shorten the transient stage dominated by single mechanism (matrix diffusion or interface), but indirectly delay the transition from the interface-dominated state to the diffusion-controlled stage. These findings not only offer new insights into the previous observations in microgravity experiments [VA Snyder, J Alkemper, PW Voorhees, Acta Mater 2001;23:699], but also theoretically describe the limiting coarsening behaviors at ultrahigh volume fractions [H Yan, KG Wang, ME Glicksman. Acta Mater 2022;117964]. [Display omitted] [ABSTRACT FROM AUTHOR]