EDIT: As i see there are smartphone apps already, which could do it also e.g. FTPsync. Or by a Filemanager on your phone that supports ftp, you can connect to the both and move the files manually.
There would be a other way. Get a RasperryPi (a Zero will work), set it up as homeserver including a ftp client. Then you can set up a automatic syncronisation of the desired folder/files, when you launch ftpd on a 3DS . But you might need write a script to meet your goal.
The easiest would be to set it up in that way, if you plug it in a usb it boot up and sets up your server in your Network (Setting it up as access point would work, too. But you will need to change the Wifi in the 3DS each time). Then it searches automatically for your 3ds ftpd server and syncronizes the folder or files. Then if finished shut it down by e.g. a additional button on the raspi.
