Neat Pic!! I just saw some on an auction site, SOLDUSA I believe it was called. I had Googled wooden primer boxes - just for kicks. There was a picture of several boxes of Federals, Fiocci, Winchester, and Remington. One of the Remingtons was a green box labeled Kleenbore, the other was a red box and it had a white triangle with Kleenbore in it. The description said it was from an Army contract and all the primers were in wooden trays. I'd say that takes it at least into the 40's and that's about when Dupont joined up with Remington. I couldn't find anything more definitive for you, like a date they started plastic. I did find that in 1960, they introduced plastic shotgun hulls. FWIW most of what I could find written with anything about 'adapted to smokeless powder' in several articles seemed to point to the 1920's. Fun Stuff.
H,mmm - Your pic shows a number on the side of the box like a lot number. You might send that and the pic to Remington and ask for some history. Bet they 'fess up! Let us know if you find out anything.
Regards,
Sweetwater