Membranes from polyether-block-polyamide (PEBA) polymers with varying chemical composition (PEBAX), cast from 1-butanol and cyclohexanol, are studied by small-angle X-ray scattering (SAXS), dry and water-swollen, and as a function of strain. The nanostructure from soft and hard domains is determined. The typical hard domain thickness is 6 nm, and the long period is 17 nm for all grades. Gas transport parameters (permeability, diffusivity, solubility) are determined for He, H-2, CO2, O-2, N-2, and CH4. Increase of polyether domain thickness leads to an increased permeability, but imperfect microphase separation contributes to gas transport properties on the same order of magnitude. Utilizing the interface distribution function (IDF) and the multidimensional chord distribution function (CDF) analysis, we find that the nanostructure of PEBA materials with a high soft block content is hardly changed by variation of casting solvent or water content. Damage of the hard domains occurs during straining. The nanostructure of PEBA materials with a soft block content close to 50% ("balanced") is a function of the casting solvent. Membranes cast from cyclohexanol exhibit a decreased hard domain fraction caused from a declined phase separation. When the hard domains become too small, early failure upon straining is observed. One of the balanced materials exhibits an irregular domain structure; the other two form lamellar stacks with a different degree of order. Swelling in water is controlled by the chemical structure of the blocks. In the PEO/PA6 copolymer both soft and hard phase swell. The swellable PTMeO/ PA12 copolymer shows growth of the soft domains only. Moderate elongation of the materials with a lamellar structure leads to orientation of the hard domain slabs parallel to the direction of strain. The extension of the slabs is < 15 nm. Further elongation induces a microfibrillar structure with increasing heterogeneity. The most probable distance of two neighboring hard domains in the microfibril along the straining direction (long period) remains constant, but the shape of the long period distribution becomes skewed. When domains are destroyed during elongation, there is a tendency to form an orthorhombic macrolattice of almost spherical fragments in addition to the microfibrils.