I always render out in layers and composit in a program like After Effects or Final Cut Studio. So, the characters would be on one layer, the backgrounds on another, and so forth.

This speeds up production in a number of ways. First, if you have a static camera shot, you only have to render out a single frame of the background and composit that in with the animated footage of the characters. Second, if you have to do multiple takes to get the motions right on the characters, or you have to change some details, you do not have to re-render the background each time. Third, the background is re-usable, so that you can use it in multiple scenes.

The same thing can be done in moving backgrounds. All you do is render out the background with a moving camera and then--AND THIS IS IMPORTANT--without changing the camera movement at all, render out the characters. When composited in After Effects or Final Cut, the result will be seamless.

I use After Effects for compositing. Open-source programs like Jashaka can also be used for compositing and I understand that Blender also has compositing functions available.