Those games are DSi games. There is no flash cart on the market - AT ALL - which can play DSi games. They call them "i" compatible flashcarts simply because it can run on the DSi, not because it plays DSi games.
Currently we have no way of running anything in DSi mode as it has a kind of encryption we cannot break. You will just have to wait.
Another issue is that the DSTTi will probably never run DSi games - even when we HAVE cracked DSi-mode - because we will probably need new hardware (not a sorftware update) in the flash carts to use this mode.
And finally: what the hell is a .dsi? We can't actually dump DSi games yet, so we definitely don't have an extension for them. The only reason those games have a dump at all is because they are just "DSi-enhanced", which means they will run on the normal DS and will do extra stuff on the DSi. But, as I said, these dumps only have the regular DS part of the rom.