A novel dynamic free-volume theory is developed to account for the influence of solvent mobility on molecular diffusion in a condensed solvent. In this theory, the diffusion process is viewed as consisting of three sequential steps of opening up a free volume, moving the diffusant into this new free volume, and filling up the position previously occupied by the diffusant. As a fundamental assumption, the time for an elementary diffusive motion is considered to be the summation of the characteristic times for the above three steps. On the basis of this and other assumptions, theoretical formulations for the diffusion coefficients in simple fluids, polymers, and interphases are derived. The model agrees qualitatively with numerous experimental findings with respect to changes of molecular diffusivity with solvent mean free-volume, temperature, and diffusant size. In the mean time, molecular-dynamics simulations of solute diffusion in a monatomic fluid and a lipid membrane are conducted to investigate the influences of solvent relaxation and solute kinetic rates on solute diffusion. The diffusion coefficient is found to depend only weakly on the kinetic velocity of solute as characterized by solute mass but change strongly with solvent mass in the simple fluid and with the chain isomerization time in the lipid bilayer. These findings are in conflict with the previous free-volume theories and the Enskog kinetic theory, but can be described satisfactorily by our present theory.