Well, for what you're trying to do, @Atrius97 basically said it all already above. GBA cart reader device for your GBA handheld to backup your saves to your PC, then mod your 3DS to have custom firmware on it so that you can install JKSM or Checkpoint, which is used to backup/restore game save files. Then, backup a Gen6/7 game (X,Y,etc. or Sun/Moon, etc.) using JKSM/Checkpoint. I then personally use a SD-to-USB USB stick to transfer the backed up save files from JKSM/Checkpoint to my PC, then can edit it with PKHeX.
Once you get to the PKHex part, move all of your Pokémon from the backed up GBA cart reader game(s) to the 3DS' Gen 6/7 backup save file in PKHeX, then export and overwrite the 3DS save with the new one once all of your Gen 3 Pokémon are on it. Put the SD card back into the 3DS then restore the Gen 6/7 sav file with JKSM/Checkpoint (whichever one you used to backup the save). From there, you'll have to install Pokémon Bank on your 3DS (which Nintendo recently said they'll end Bank support within a year or something like that) and move the Pokémon from Bank to Home by downloading the Home app on your mobile device. Just make sure you purchase the Pokémon Home premium plan.