A membrane cartridge is manufactured by repeatedly folding and joining two strips of membrane to foam a cross-pleated cartridge with a stack of openings or fluid passageways configured in an alternating cross-flow arrangement. The cartridge can be modified for other flow configurations including co-flow and counter-flow arrangements. Methods for manufacturing such cross-pleated membrane cartridges, as well as apparatus used in the manufacturing process are described. Cross-pleated membrane cartridges comprising water-permeable membranes can be used in a variety of applications, including in heat and water vapor exchangers. In particular they can be incorporated into energy recovery ventilators (ERVs) for exchanging heat and water vapor between air streams being directed into and out of buildings.