Downtube shifters won't stay in low gear on rear derailleur
Downtube shifters are usually called "friction shifters" and friction is key when fixing the issue you describe. I own and periodically use an old Cannondale with such system. Every now and then I find myself with the same problem as yours. How to fix it? Easy, just tighten the lever of the shifter. 

Mine came with an integrated metal triangle handle (couldn't find its exact name, but I've attached a picture as an example.) So they can be tightened using your bare fingers and on the fly, but some other models might need to use a screwdriver. The H and L limits of the derailleurs will have nothing to do with this problem.