DSiWare purchases are non-transferrable
Unless you break your DSi
Nintendo has confirmed that any DSiWare games or applications bought for a DSi can not be transferred onto another machine.
In other words, the content is locked to the handheld it was downloaded onto. So if you're planning on buying a different coloured DSi – and the new batch of metallic blue, light blue and red due on October 23rd is tempting – and want to enjoy the same content, you'll have to pay for all of your purchases again.
There is a way around this, though. Nintendo will transfer the content from a broken DSi when it (along with its replacement) is sent in to their offices.
Nintendo Life made the discovery by getting Nintendo Australia to reveal its DSiWare policy and this has since been confirmed by Nintendo's other territories.