Put off packing their stuff. Put off packing up the majority of your child's toys and clothing until as late as you're comfortable with. We do our entire kitchen before we do her toys. Then make sure their room gets put together almost first when you arrive at your new home.
Have new fun stuff ready (especially if there's a long road trip involved). These can be simple things like stickers, a small doll, a notebook and crayons, etc. Obviously you don't want them to take up too much space, but they are lifesavers when your kid is going crazy with that horrible mixture of boredom and messed up routine. Dole them out sparingly.
Carve out a space for them. I mean physically, make sure there is a space for them to play right up until the move. Give them their own suitcase for their clothes and their own backpack for the toys they want to bring. Give them some ownership over their stuff and a little control over the overwhelmingly out-of-control situation they are in.
Carve out space for their feelings. It's hard to manage your own emotions during a move, let alone someone else's. But toddlers and preschoolers need you to be aware, especially during stressful times like moving, that when they are acting out, it's not personal. They're stressed, and they need you to help them with that. Be there to listen. Ask open-ended questions to let them talk about things that have been bothering them or that they've been wondering or worrying about. Keep routines as similar as possible to make them feel more secure. Hug, snuggle, and tickle often. It will be hard, but try to carve out time to play with them.
Let them help. Find ways to let them help with the packing, even if it's only having them hand you things to put in a box. They can also decorate the moving boxes with stickers and drawings, or fetch things from another room. This is another good way to give them some control over the situation. Last year, Lucy was very insistent that she be able to put something onto the truck. So we let her haul one of her little chairs up the ramp and onto the truck. Work stopped until she was done, because it was important to her. Somewhat related -- don't pack up things that are important to them while they are asleep or away. It will help them to see their things go into the boxes, so it's not as if their things have just disappeared.
Say goodbye to things. Be aware when you are on your last visit to a place you go often -- a favorite park, the library, grocery store, a friend's house, etc. Let them know beforehand that this will likely be the last visit, then say goodbye to the place as you leave. Sometimes it seems like very young kids won't know the difference, but with my own daughter I've seen how giving her information that I don't expect her to be able to fully process has helped in many situations. (This started when I decided to start letting her know when I was going to the bathroom when she was around 4 months old, rather than just setting her down, going, and coming back. It sounds crazy, but it helped her to stay calm. Since then, I've pretty much operated on the assumption that more heads up and more information about what will happen is always better.)
If your move involves a road trip -- all the usual road trip advice applies. Plan for lots of stops, pack lots of snacks, don't be afraid to use a DVD player to keep them content. I've been adding road trip tips to this Pinterest board, starting with the busy bags I chronicled on this blog.
The only other road trip advice I've got is, especially for the very littles, they must run around at stops. There are many ways to achieve this, most notably, stop at a fast food place with a playground, or stop at a highway rest stop that has a large grassy area. You can bring bubbles and have them chase the bubbles. One of our favorite ways to get moving at a rest stop is to play "olly olly oop", which is a game Lucy learned from Blue's Clues. Basically, someone yells "olly olly oop, [physical object]", then everyone jumps up, then runs and touches the object. So at a highway rest stop, it would go something like "Olly olly oop, trash can! Olly olly oop, that tree! Olly olly oop, picnic table!" and so on until everyone is out of breath. Very effective. Kids who are a little bit older will be able to play tag.
So I guess this all basically comes down to remembering to remember all of your kid's needs. During a move, sometimes it seems like all you'll be able to manage is feeding them and making sure they sleep, but humans need more! If you can anticipate their needs, and the whole experience will go more smoothly.