Moving, of itself, is not a frugal thing to do, unless maybe you pack everything into free grocery store boxes (if you can find any) and stuff it all into the back of your car. (Or borrow Grandpa's hauling-stuff trailer, as my parents used to do.) The last time we moved, it was with a station wagon and the help of friends. This time around, we went with a moving company for the furniture and heaviest boxes, which did not save money but did save time, aggravation, and thrown-out backs. Because we had early access to the new place, we did pre-move what we could manage by ourselves.
The frugal part of moving, for us, was taking the opportunity to size down, to sell and donate (and trash) the things we no longer needed, and to make better use of what remained. You've probably all had the experience of moving a somewhat-ignored picture or a piece of furniture to a different spot, and all of a sudden you notice it again? Moving house is like that, only multiplied.
Scaling down does not have to mean bare walls and futons. It doesn't mean having an apartment that looks like nobody lives there.
Or sticking the kitchen scissors on a round fridge magnet. (Only a very strong one will work, though.)
Or finding a home for kitchen clothespins in what used to be a magnetic laundry-room stray-button/coin catcher.
Or resurrecting the grandparents' kitchen table, which had gone to the workshop when we bought the green table which is now in our dining room.
Or putting the guinea pig in the centre of life. (Muffin says the meal service is much improved. He is already tuned in to the sound of the fridge door opening.)