Many crystal plasticity models can predict evolution of crystallographic texture along a deformation path imposed. It is quite straightforward to derive a homogenized stress response of the material upon a strain rate stimulus, but not vice-versa. In this work we propose an iterative scheme that transforms a strain-rate driven CP model into a stress-driven one. This approach enables us to study the evolution of the material state (such as crystallographic texture) that results from an arbitrarily chosen stress path. Furthermore, we are able to track quantities that can be determined from the state, e.g. plastic anisotropy. The proposed analysis of the stress response consists of two steps. Firstly, an efficient search procedure is employed to determine a strain rate tensor corresponding to the superimposed stress, which involves very few evaluations of the CP model. Once the strain rate is known, the CP can predict an update of the material state. These steps can be repeated for a predefined sequence of loadings. We consider two homogenization schemes: the Full Constraints Taylor and the ALAMEL. The latter not only takes into account interactions between the crystals in the material, but also handles stress equilibrium at the grain interfaces. We demonstrate the development of textures and anisotoropy for typical fcc and bcc materials upon monotonic and non-monotonic loading. The predictions are verified by mechanical uniaxial tensile tests, followed by texture measurements at intermediate deformations. The experimental setup allows tracing the plastic anisotropy in terms of instantaneous and cumulative r-values. It is shown that the stress-driven CP modeling is able to correctly predict the material flow and the developed texture. Finally, the origins of discrepancies are discussed and measures to counteract these are proposed.