A harvesting machine includes a harvesting header with a header frame, a flexible cutterbar assembly attached to the header frame along the length thereof and configured to cut a crop, and a conveyor assembly positioned behind the cutterbar assembly and operable to receive severed crop material from the cutterbar assembly. The conveyor assembly includes a fore-and-aft draper with a draper frame swingably supported on the header frame at a pivot where the draper frame projects forwardly from the pivot, such that the draper frame is operable to swing relative to the header frame. The fore-and-aft draper includes a draper belt supported on the draper frame to present a rearwardly moving run for conveying the crop rearwardly.