Multi-structural polymer composites are widely used in the mechanical engineering, automotive, aviation and oil refining industries, as well as in the printing industry as a shock-absorbing deckle on the offset cylinders of printing machines. During offset printing, composites come into contact with inks and washing solutions, the components of which penetrate the material and cause the polymers to swell. This process degrades the print quality, and for this reason the study of its features is relevant. The prerequisites for this work are the study of the fundamental laws of diffusion and sorption of liquids by polymers with different micro- and macro-structures in different physical states and in different forms (e.g., films, sheets, fibers and fabrics). The combination of polymer materials in the composition of multi-structural fabric blankets makes it possible to obtain materials with unique mechanical properties and high resistance to liquid penetrating media and to use them in high-tech processes of multi-color printing with high resolution and color rendering. This article reports for the first time the kinetics and thermodynamics results obtained from the swelling of multi-structural polymeric blankets in solvents used in printing, and the effect of sorption of different polar liquids on the viscoelastic strain under compression during the operation of the damping systems of printing machines. Using mathematical models of activated liquid diffusion in polymers and deformation of a viscoelastic body, the swelling rate constants, solvent diffusion coefficients (the kinetic characteristics of the swelling process) and Flory–Huggins parameters (the thermodynamic characteristics of the interaction of the solvent with the composite) for composite–solvent systems with several chemical composition variants were determined. The elastic modulus and the viscosity coefficient of the composite under liquid saturation were calculated based on the experimental cyclic compression data. The range of change in the compression and restoration times of the polymeric blankets (0.09 s ÷ 0.78 s) was determined. It was shown that the composite swelled to a limited extent in all the studied liquids. All solvents used were thermodynamically poor (χ > 0.5). It has been established that rubber–fabric blankets coated with nitrile rubber are the least resistant to the action of dichloroethane, and that blankets with layers of polyolefins are not resistant to ethyl acetate. Water significantly affects the physicochemical properties of rubber–fabric blankets with a large proportion of cotton fabric layers. The data obtained can serve as a basis for optimizing the compositions of inks and cleaning solutions, as well as a theoretical basis for the thermodynamics of composite–solvent systems.