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We address the challenging open problem of blindly separating time/position varying mixtures, and attempt to separate the sources from such mixtures without having prior information about the sources or the mixing system. Unlike studies concerning instantaneous or convolutive mixtures, we assume that the mixing system (medium) is varying in time/position. Attempts to solve this problem have mostly utilized, so far, online algorithms based on tracking the mixing system by methods previously developed for the instantaneous or convolutive mixtures. In contrast with these attempts, we develop a unified approach in the form of staged sparse component analysis (SSCA). Accordingly, we assume that the sources are either sparse or can be “sparsified.” In the first stage, we estimate the filters of the mixing system, based on the scatter plot of the sparse mixtures' data, using a proper clustering and curve/surface fitting. In the second stage, the mixing system is inverted, yielding the estimated sources. We use the SSCA approach for solving three types of mixtures: time/position varying instantaneous mixtures, single-path mixtures, and multipath mixtures. Real-life scenarios and simulated mixtures are used to demonstrate the performance of our approach.