Do you recommend continuing to move the Eglu through the winter?

You can leave the coop in one place, as long as they have perches (so they can be above the cold ground if they want) and other toys to interest them, but you can also move it around if you prefer. 

Some customers have their coop installed inside runs or attached to them, making them much harder to move, so if you did want to keep them in one place, this would be fine.