This article proposes a unifying design framework for dynamic feedback controllers that track solution trajectories of time-varying generalized equations, such as local minimizers of nonlinear programs or competitive equilibria (e.g., Nash) of noncooperative games. Inspired by the feedback optimization paradigm, the core idea of the proposed approach is to repurpose classic iterative algorithms for solving generalized equations (e.g., Josephy-Newton, forward-backward splitting) as dynamic feedback controllers by integrating online measurements of the continuous-time nonlinear plant. Sufficient conditions for closed-loop stability and robustness of the algorithm-plant cyber-physical interconnection are derived in a sampled-data setting by combining and tailoring results from (monotone) operator, fixed-point, and nonlinear systems theory. Numerical simulations on smart building automation and competitive supply chain management are presented to support the theoretical findings.