One reason why standard formulations of the central limit theorems are not applicable in high-dimensional and nonstationary regimes is the lack of a suitable limit object. Instead, distributional approximations can be used, where the approximating object is not constant, but a sequence as well. We extend Gaussian approximation results for the partial sum process by allowing each summand to be multiplied by a data-dependent matrix. The results allow for serial dependence of the data and for high dimensionality of both the data and the multipliers. In the finite-dimensional and locally stationary setting, we obtain a functional central limit theorem as a direct consequence. An application to sequential testing in nonstationary environments is described.