Using a hybrid code, we examine the effects of mildly oblique, low-frequency Alfv¿n waves on cross-field pressure-balanced structures of varying scales. We show that the evolution is organized by a parameter ξ = AwcAk$perp$/ω, where Aw is the relative wave amplitude, k$perp$ is the characteristic wave number of a cross-field structure, and ω is a characteristic wave frequency. This parameter is a measure of the relative displacement of the structure by the wave. When ξ $ll$ 1, agreement with small-amplitude solutions and linear wave theory is good. Waves can refract and undergo cross-field energy transfer, which is a finite wavelength effect. When ξ is large, advection of the structure is significant and small-amplitude solutions are not valid. The rate of wave refraction diminishes as the effective Alfv¿n speed gradient is diluted over a wave period by significant advection. When the structure is purely magnetic and polarized like an Alfv¿n wave, significant advection converts the structure into waves which have an Alfv¿nic character but do not always satisfy energy equipartition. Nonlinear wave interactions occur between Alfv¿n and gradient-derived waves, wherein the Alfv¿n wave undergoes partial reflection and continued cross-field energy transfer. When sufficiently small parallel scales are produced by advection, proton cyclotron resonant heating and damping occurs on all field lines so that both Alfv¿n and gradient-derived waves undergo dissipation.