What's the fastest way to move builds from world to world?

The WorldEdit solution is pretty easy, since WorldEdit supports mcedit schematics and we can make use of these...

1. //copy (as you would usually do)
2. //schem save mcedit <name> ( I used "bunny" here, no extension needed)

3. Locate the Schematic file in your WorldEdit/schematics directory. It should have the name you gave it plus the .schematic extension.
4. Copy that file over to the same directory of the target world
5. //schem load mcedit <name> will load it into the clipboard

6. //paste

Remember: Just like with normal c&p it will remember your position when you copied it and paste it relatively to your position when using //paste. Make sure to stand on the right position when pasting it. Standing on the ground right next to the structure when copying it is usually a good idea.

dly's answer does work but if you want to move builds without using world edit, another way is to use structure blocks