Rigging - manipulating a rigged model

I have a model which seems to work fairly enough. I can pull it into another file and add a proxy to get at the armature, which allows me to pose the thing;

What seems to be the problem is what happens if I want to move (ie, translate, rotate or scale) the model as a whole, after I’ve created the proxy. What seems to happen is that the mesh moves and leaves the armature behind.

Also, if I create another instance of the group (ie, I want two of the same character in-scene) the second one appears in the first one’s pose - and I have the same problem if I make a proxy to change it, where the mesh moves away but the armature doesn’t.

Tangeantial question: In what I suspect is a bad, hacky attempt at a fix, I tried to parent every free-floating bone in the rig to a base bone, which I could at least use to position and move the character around with. Unfortunately, when I try to move this bone, the rig itself goes haywire and the base bone won’t move where I put it, and starts rotating in odd directions instead.

What’s likely to be causing this, and how can I stop it without drastically reworking my rig?