You should be able to go back and forth between Ghosts on the 360 and One with out any problems. I personally haven't had any trouble. The console switching and Ghosts progression between consoles has been seamless.
Are you signed out of the One when you boot up the 360?
I dont know about the stats though, but the DLC YES. I am not sure but I thought I read that the upgrade to ONE was a one time deal and not backwards.
If you're asking about your multiplayer levels, then yes - they switch between consoles, I switch between consoles on a regular basis to party up with different friends and have never had a problem - much like iHattoriHanzo0. Your stats do not transfer between consoles and the same goes for any campaign progress.
You may want to double check you've selected the correct soldier, if that is not the problem you could try contacting
https://twitter.com/ATVIAssist for help.
They're independent - your K/D and such on the 360 remain what you left them as, your K/D on the One is like a separate copy made when you first made the transfer. Those stats will go up and down according to how you play on the One, but won't affect the 360 stats, and vice-versa. I'm not exactly sure why they did it that way, or why it had to be done that way, it would make more sense if ALL of your Barracks data remained updated and synched for both versions, but yeah.. lol can't get the icing with your cake always.. well in this case you seem to be missing half your cake as well.. but I digress.