This work presents a general framework for designing arbitrary high-order well-balanced discontinuous Galerkin (DG) methods for hyperbolic balance laws with various equilibrium states. Unlike traditional approaches that focus solely on hydrostatic equilibrium, our scheme also exactly preserves moving equilibrium states for models such as the shallow water equations, the Ripa model, and the compressible Euler equations under gravity. The key innovation is approximating equilibrium variables within the DG piecewise polynomial space instead of the conservative variables, which is crucial for achieving the well-balanced property. Our method offers flexibility by being compatible with any consistent numerical flux and eliminates the need for reference equilibrium state recovery or specialized source term treatments. Furthermore, this approach facilitates the development of well-balanced schemes for non-hydrostatic equilibria, including two-layer shallow water equations and rotating shallow water equations. Extensive numerical tests, including cases with moving and isobaric equilibria, demonstrate both the high-order accuracy and exact preservation of equilibria across various flow scenarios governed by hyperbolic balance laws.