Unlike Xbox 360, PS3 and PSP there's no method for transferring the rights to downloaded software from one machine to another by the end user. In fact, Nintendo Australia informed the site that, should a DSi break and its owner want its DSiWare moved to a new unit, both must be mailed in to Nintendo, which will perform the process.
We've asked Nintendo of America if it has a similar policy for what should be a matter of a simple phone call to de-authorize one DSi and authorize another to download content you've already paid for. We're anticipating a "yes" in response. Then again, as GameSpot points out, NOA's end user license states "software downloaded from Nintendo DSi Shop is licensed to you, not sold." Pays to read the fine print.