I would suggest that they need enough water to be properly wet (not merely damp), but not enough to slosh. Definitely easier with RC due to its greater wet rigidity.

Maybe if you had some arrangement of clips in the bottom of the tray that could hold each corner of the print so that no matter how violently you drive, the prints won't end up all wadded up at one end of your container. Or you could put flat plastic on top of your prints and clamp that down loosely with the container lid; it'd prevent all movement of the prints.