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The objective of the paper is that of constructing finite Gaussian mixture approximations to analytically intractable density kernels. The proposed method is adaptive in that terms are added one at the time and the mixture is fully re-optimized at each step using a distance measure that approximates the corresponding importance sampling variance. All functions of interest are evaluated under Gaussian product rules. Since product rules suffer from an obvious curse of dimensionality, the proposed algorithm as presented is only applicable to models whose non-linear and/or non-Gaussian subspace is of dimension up to three. Extensions to higher-dimensional applications would require the use of sparse grids, as discussed in the paper. Examples include a sequential (filtering) evaluation of the likelihood function of a stochastic volatility model where all relevant densities (filtering, predictive and likelihood) are closely approximated by mixtures.