First of all, game save files are stored on the SD card in the Nintendo 3DS folder. The only saves on the NAND are for things like Activity Log and Mii Plaza.
Second of all, the contents of the Nintendo 3DS folder are encrypted per console. This means you can't just move an SD card to another console and expect all the games to show up.
Third of all, this is how the transfer works:
1. Nintendo's servers change their record of which console the NNID belongs to (if any NNID is present). This means you can sign into the NNID on the new console and no longer sign in on the old one.
2. This is the important part: The encryption on the Nintendo 3DS folder is changed so that the new console can read it. If your data is not present on the old console at the time of the system transfer, it will miss out on being converted for the new console. So, if you move the data before the transfer, then no console will be able to read it!! And your save data will be lost. Except if you restore the pre-transfer NAND backup, then the old console can read it, and you could recover the save with JKSM and inject it on your new console.
I know this isn't exactly right, because even when you do a transfer properly, you can restore a backup on the old console and both consoles can read the data. I think maybe the data is re-encrypted so both consoles can read it, but then of course the old console gets formatted and can't read it anymore, so it has the effect of changing it so only the new console can read it.