Yes, there is a way to play old Game Boy Advance games on your new Nintendo DSi. You just have to download the GBA DSi Emulator. Yes, it is possible to play GameBoy Advance game backups (aka GBA ROMs) on your DSi. You have to use a special DSi flash card adapter though, for MicroSD cards. Then use the emulator and place the GBA games on the Micro SD card. This is the surefire way to download DSi games on your SD card.
Have an iPhone? Check out all 200+ new features coming in iOS 13.