Intracellular dynamics in living tissue are dominated by active transport driven by bioenergetic processes far from thermal equilibrium. Intracellular constituents typically execute persistent walks. In the limit of long mean free paths, the persistent walks are ballistic, exhibiting a "Doppler edge" in light scattering fluctuation spectra. At shorter transport lengths, the fluctuations are described by lifetime-broadened Doppler spectra. Dynamic light scattering from transport in the ballistic, diffusive, or the crossover regimes is derived analytically, including the derivation of autocorrelation functions through a driven damped harmonic oscillator analog for light scattering from persistent walks. The theory is validated through Monte Carlo simulations. Experimental evidence for the Doppler edge in three-dimensional (3D) living tissue is obtained using biodynamic imaging based on low-coherence interferometry and digital holography.