Flexible packaging materials consisting of several layers of very dissimilar materials are currently considered for replacement by multilayer polyolefin films as well as functional flexible paper materials to improve recyclability. Such materials however lead to new challenges in processing on existing packaging machinery. This paper focuses on the sealing process of such materials, especially looking at pillow pouches and the challenges occurring at the junction between cross and fin seals where the total thickness jumps from two to four layers. It is shown that the application of conventional machinery and sealing tools does not lead to a gastight seal within the operational sealing window of such materials. The origin of the permanent failure is the lack of material fraction capable to flow as a viscous melt into the gap at the fin seal. The limited compressibility of the nonmelting materials and their bending stiffness tend to increase the gap. In case of paper, a flexible sealing jaw, equipped with an elastic insert, enables a focused application of pressure at the layer jump of a poorly compressible material. This results in clearly improved gas tightness of all selected materials in the range of the used test methods (leak size 10 μm). The leakage rate of papers was reduced from 3–4 mbar·l/s with rigid tools to 1 mbar·l/s with flexible tools. The leakage of 2 mbar·l/s with OPP/CPP material could be prevented and no leaks were detected. With OPP/PE, leak-proof results were obtain within reduced temperature at 140°, which is near the level of acceptable seal quality with low shrinkage effect.