You need a way to dump the save from the cart and then write it back to it every time you want to make edits, of course. If it's just individual Pokémon, you can edit it in your flashcard save and then trade it, but that requires two consoles. If you want to use those Pokémon in later games and/or online, you should make sure to keep them legal (lots of tutorials here regarding that).
As for the main topic:
There's a Wi-Fi Backup tool. I've never used it before, but it should work. It requires a DS Phat or Lite and a Slot-1 card only. If yours is a DSi, then you need to install CFW. https://gbatemp.net/threads/nds-backup-tool-wifi.252623/