What happens when you access the DSi System Apps on a Nintendo 3DS? 

You can access these apps by downloading off hshop and unpacking the .cias with FBI, or you can download them directly off hshop on the 3DS itself.
Findings: Every single app works, with the exception of the demonstration app. The DSi Shop will open, but will not load the shop itself. The 3DS Transfer Tool will also open, but give an error upon startup. The System Settings app will only open if you have the right files in the TWL NAND partition. The latter three apps may not work at all on New systems, but will always work on Old systems. (Hence why I used my phone to record the ones on my 2DS, as my New 3DS XL is the one with a capture card)
(Note, I used an emulator to record the DSi portions of the video, but I can confirm they all work.)