Nintendo has explained to Eurogamer that DSiWare purchases cannot be transferred to additional DSi machines, meaning any games or applications are locked to whichever system they're bought on.
The explanation comes as Nintendo prepares three new DSi colours for Europe: Metallic Blue, Light Blue and Red. They arrive on 23rd October, pre-installed with Facebook and Flipnote Studio.
Nevertheless, Nintendo is prepare to transfer DSiWare downloads should your DSi break and you send it in for repair or replacement. That is the only way.