An apparatus and process (20) for forming an embossed web or other article can include moving a target, composite web (26) along an appointed machine-direction (22) at a selected web speed, and operatively contacting the target web (26) with a rotary embossing device (38) to thereby form a non-linear embossment region (82) in at least a selected embossment portion of the target web (26). For example, the web speed can be at least a minimum of about 1.9 m/sec. In a particular aspect, the embossing device (38) can include an outer peripheral surface (42) having a lateral cross-direction (24) and a circumferential-direction (40), and can include a non-linear embossing member (44) located on the outer surface (42). In another aspect, the embossing member (44) can be configured to include a selected traversing occurrence along the circumferential-direction (40) of the embossing device (38). In a further aspect, the embossing member (44) can be configured to include a selected lateral traversing distance (50).