personally kept all my doll boxes. When I was selling some of my tonners, it was very easy to ship them out. When I am buying a doll, I would like the box, but its not a deal breaker. Some of my tonners did not come in their original boxes. I have a special shelf where I store all of mine in the basement. I even have the shippers but those are broken down flat. It does take up space but if you are thinking of downsizing, it would be useful to have the boxes. I don't think the doll's value means as much with/or without the box, as I think the condition of the doll itself is more important.
Some collectors are picky though and want the box. Best of luck with your move!