We present a generalized variational framework for computational homogenization posed on a six-dimensional product space, which recovers classical FE^2 and IGA^2 schemes as special cases for suitable discretizations and solution strategies (see [1]). The formulation provides a systematic basis to analyze conventional approaches and to improve global convergence by adopting a fully monolithic solution strategy. In particular, we introduce a null-space reduction that eliminates local RVE related calculations and significantly reduces the number of unknowns in the higher-dimensional problem. As a result, consistent tangent operators follow directly from the variational structure, avoiding nonstandard linearization techniques. The framework is readily extended to higher-order continua of arbitrary order n following the ideas in [2]. Moreover, we adapted the numerical framework on inelastic constitutive behavior (e.g. visco-elasticiy), where the monolithic, variational setting also improves robustness and computational efficiency compared to conventional strategies. We discuss classical and variationally consistent formulations in detail and assess their performance by comparison with reference solutions including analytical solution of a benchmark problem. Finally, we demonstrate the applicability of the proposed method for general multiscale examples.