...from one computer to another? I can't find any map names or folders that look relevant.

I want to do this because EDCE doesn't work with W7 lappy, I installed it on W10 computer, now I am upgrading lappy to W10 so I can play on it. I uploaded my map & may have to wait for approval to download it on the lappy but there must be a better way.

If you uploaded your map then it is available for you to download.

If you want the actual map file, look in the maps directory in the maps directory of the <Game Data Directory>. See the admin screen to get that location. 

The manual also has information on this.