Most likely the mission was already in the rom somewhere and the "download" just unlocks it or something similar to how you can download player created missions in Mario vs. DK 2 or Scribblenauts.

It'll change the save file, not the game itself, since the DS doesn't write directly to .nds files.

 

The Download Mission is stored in the save file so you can download it on a DS and then play it on an emulator. using the save file from your DS