It depends on how cheap the 360 is. You'll still be able to get games on that for quite a while. The kinect should work for your kids. However, if it isn't significantly cheaper then go ahead and get the xb1. It has a better kinect plus other features a non gamer can enjoy and use. It will be good for a decade or so. But right now there might still be certain problems with the system that haven't been ironed out or even discovered, there are limited games for it at this point. It just depends on how much cheaper the 360 is I think. I don't know for sure, and I'm just going by what I've read and have seen on the web. I have a 360 and not a XB1. I'm probably going to wait a year before getting a XB1 or PS4. I'm still not sure which one I'll get.
I would personally recommend the Xbox One. I believe that Kinect 2.0 is simply that much more improved, and as this is mainly for your 3 kids and their use of the Kinect, it becomes a no-brainer for me. I haven't been that big on Kinect yet, but the first time I hooked up the Xbox One and saw how clear Kinect 2.0 was, I was very impressed. Honestly, I haven't played any Kinect-based games though, so I can't attest to how great it is in action. But I love that it instantly recognizes me when I turn on the system and loads up my gamer profile. I think it would be pretty cool as your kids are taking turns playing a game, the new Kinect can instantly identify each of them, and will know who is playing.
