<p>The breathing behavior of Cosh-Gaussian laser beams is examined in the context of cubic-quintic nonlinear media, where the quintic nonlinearity is modulated along the propagation axis. Using the method of moments, evolution equations are derived to describe the beam's spatial dynamics. The impact of this axial modulation on beam stability and oscillatory behavior is analyzed, revealing that controlled variation of the quintic term enables tunable breathing patterns. These modulated dynamics offer a flexible mechanism for managing beam propagation in nonlinear environments, with potential applications in laser beam shaping, optical filament control, and the design of nonlinear photonic devices.</p>