In this work, we derive a generalized variational framework for computational homogenization, which recovers traditional FE² and IGA² approaches when specific discretization and solution techniques are applied (see Hesch et al. [1]). This framework enables a detailed analysis of classical FE² schemes, enhances global convergence properties through the selection of suitable shape functions, and introduces a null-space reduction method to systematically decrease the number of unknowns. These advancements significantly simplify the construction of consistent linearizations, even for higher-order problems. The effectiveness of the proposed approach is demonstrated by comparing results with its analytical solution of a one-dimensional benchmark problem at the macroscale and with a one-dimensional microscale representative volume elements (RVEs). Additionally, the method's applicability is showcased through the two-dimensional Cook’s membrane problem. Finally, we outline the potential for extending this framework to higher-order continua of the n-th grade, following Schmidt et al. [2]. This extension would incorporate additional terms from the Taylor approximation within the RVE to account for higher-order macroscale kinematics, offering a promising avenue for future research.