I have a DSTWO, which totally works on DS, DSLite, DSI, DSIXL, 3DS, and 3DSXL.
However, on the 3DS it only works in DS mode (they still haven't cracked them to play 3DS carts/software).
I also have an R4DS but it won't load on my 3DS (it doesn't recognize the cart).
So you might want to invest in a DSTWO or higher flashcart.