A cushioning conversion machine (20) including a conversion assembly (30) which converts a continuous sheet of multi-ply stock material (22) into a three dimensional cushioning product (32) and a stock supply assembly (550) which supplies the multi-ply stock material (22) to the conversion assembly (30). The conversion assembly (30) includes a forming assembly (52) which inwardly turns lateral portions of the stock material (22) as the stock material (22) travels therethrough in a downstream direction. The stock supply assembly (550) includes a separating device (574) and/or constant-entry device (580) designed to minimize tearing due to excessive tension in the lateral regions of the stock material (22). Additionally or alternatively, the forming assembly (52) includes a shaping member (590) having protrusions in its turning perimeter and/or holding surface to modify the shape of the cushioning product. Another embodiment of stock supply assembly (350) includes a stock-dispensing device (370) which dispenses the multi-ply stock material (22) from a continuous sheet thereof and a transversely extending member which is positioned to engage at least one of the plies of the dispensed multi-ply stock material (22). The transversely extending member has end portions which are inwardly tapered relative to the central portions of the member and towards the adjacent ends of the member. The inwardly tapered lateral end portions being positioned to engage the ends of the lateral portions of the at least one ply of the stock material (22).