A composite structure suitable for making transdermal delivery devices includes in sequence: (a) a liner film layer; (b) a containment layer having from zero to 15 wt. % content of vinyl acetate repeating units, the layer including either a thermal bonding copolyester resin or a combination of an ethylene-vinyl acetate component and a nonpolar polymer; and (c) a polyester film layer less than 20 μm in thickness. A drug layer lies either between the liner film layer and the containment layer or within a concave depression in the containment layer. A composition including an ethylene-vinyl acetate component and an ABA block copolymer, wherein the A segments of the block copolymer are styrenic segments, is also provided. The composition includes between 1 and 15 wt. % of vinyl acetate repeating units, based on the total weight of composition exclusive of any solvents.