An apparatus for making an elastic laminate including a crosslinked elastic layer formed of an elastic copolymer includes a die for extruding a layer of crosslinkable elastic copolymer, at least one roll for conveying the elastic layer, a supply source for providing at least one nonwoven web, a lamination nip for laminating at least one nonwoven web to the elastic layer, a processing unit for crosslinking the elastic layer and a winder for collecting the elastic laminate. The crosslinkable elastic copolymer may be a crosslinkable elastic styrenic block copolymer, a crosslinkable semi-crystalline polyolefin plastomer, or a crosslinkable propylene-ethylene copolymer. The crosslinkable elastic layer may be extruded in the form of a plurality of parallel filaments such as by a filament die. The processing unit may be positioned between the at least one roll and the lamination nip or may be positioned between the lamination nip and the winder. The processing unit may have an open or a closed configuration and may be an electron beam processing unit.