First you do the transfer, then after the transfer process is over, turn off the 3DS you transfered to, and insert the old SD card. Then turn it back on.
Look this up just to make sure, but if I remember correctly, for an Old to New 3DS transfer, you do the transfer, then you go into the system menu. I think under data management, there's an option to manage your New 3DS's SD card over wifi, so you set a password on your 3DS, then access the files on your computer, and just copy + paste the old SD card stuff onto the new 3DS. Double check to make sure though, I haven't done this in like 2 years!