Simple Guide to Making Cartridge Backups

Before starting, please know I take no responsibility if you bork something and break your system. It should not be possible, but if you manage to break something, I will not be able, or obligated, to help you.

Prerequisites
This guide assumes that you are using a 3DS with a version  no higher than  9.2.0-20, and that you are able to access the Homebrew Launcher, rxTools, FBI and the Gateway 3DS Menu. If you are on version 10.3.0-28, downgrade using sysUpdater using this guide . If you are on a compatible version, but do not know how to access rxTools or Gateway 3ds, follow this excellent guide .

What You Need
* I do not own a New 3DS, and cannot attest to whether or not this guide will work properly on a New 3DS. It should, but don't blame me if it doesn't.
 * A New* or Old 3DS on version number 9.2.0-20
 * Access to the Homebrew Launcher, rxTools and the Gateway 3DS Menu
 * Access to a Windows Machine
 * This 3DS to CIA converter
 * A homebrew utility called svdt, available here
 * Adequite space on your SD card. You should have  at least  two to three gigabytes available.

Tutorial

 * 1) Select your cartridge that you would like to back up. For this tutorial, I will be using Pokemon Alpha Sapphire.
 * 2) Go to the Homebrew Launcher and start svdt. It will have you select a title to extract the save data from, you should pick whichever game matches your cartridge.
 * 3) The program will start, and when it is finished, the bottom screen should have a message in green text reading, "Standard dump to SD was invoked." Your save data has now been extracted from the cartridge, and saved to your SD card. You do not have to do this step now, but it will make it easier in the long run.
 * 4) Exit svdt and the Homebrew Launcher, and go to the Gateway 3DS menu. You cannot get into this menu from emuNAND, you MUST access it from your sysNAND. Once in the Gateway 3DS menu, scroll to the right until "Backup 3DS Game Cartridge" is selected. Hit the "a" button to select this option, and then hit the "start" button to begin dumping the game. Take note of the last four digits of the game code. This will take some time, especially if your game is very large. AS took about 20 minutes for me. Be patient.
 * 5) Once Gateway 3DS tells you that it has finished, press the "a" button to exit, shutdown your system, and transfer your SD card from your 3DS to your PC. If you haven't, extract the 3DS to CIA converter you should have downloaded into it's own folder.
 * 6) On the root of your SD card, there should now be a file with a .3DS extension, whose name should match the four letter code given before. Copy this file from your SD card to the "roms" folder in folder your extracted the converter into. You can now delete the .3DS dump from your SD card, in order to free up more space.
 * 7) Next, start the 3ds_simple_cia.exe executable. On the window that pops up, click the "Create 'ncchinfo.bin' file" button. A window will pop up asking you to select the "roms" folder that you extracted your .3DS dump to. It should be selected by default, but double check. Click "Select Folder", and a new window will pop up. Navigate to the root of your SD card, that the name of the file is "ncchinfo.bin", and click "Save". Verify the the ncchinfo.bin file is present on the root of your SD card. Once you have done this, eject your SD card and place it back in your 3DS.
 * 8) Boot into the rxTools menu, and navigate to "Decryption". Select this, and scroll down to "Generate Xorpads" and select that. It will begin to generate your xorpad files, needed to convert the title to .CIA format. When it is done it may give you an error reading "Error opening SDinfo.bin", that is okay. Continue, shut down your system, and transfer your SD card back over to your PC.
 * 9) On the root of your SD cards, there should be two new files, both with an extension of ".xorpad". Copy these files to the folder in the converter folder called "xorpads".
 * 10) Again, start the 3ds_simple_cia.exe executable. This time, click the "Covert 3DS ROM to CIA" button. A folder selection window will open, and again, make sure it has selected the "roms" folder. The default option should still be correct. Click "Select Folder", and the program will start. It will appear to freeze, but do not close it! It only appears this way during the conversion process. When it is done, a window will open telling you so. NEW 3DS ONLY:  If you are using a New 3DS, or your emuNAND is older than the most recent firmware, then check the "FW Spoof" box and keep the default "2E02" value. This will allow you to run games who want a newer firmware than your emuNAND is running without issue, by tricking the game into thinking it wants 9.0 or newer.
 * 11) Open the "cia" folder, and in this folder should be a file with a .cia extension, whose name should match the four letter code given before. Feel free to rename this. Copy this file over to your SD card. Unlike other steps, this does not have to be in the root of the SD card. I like to make a folder where I put my .cia files to keep my SD more organized.
 * 12) Place your SD card back into your 3DS, boot into your emuNAND, and launch FBI. On the bottom screen, navigate to where you placed your generated .cia file, and install it. Make sure your are installing to SD (the default option). You could damage your device if you install the cia to your NAND. This process will also take some time, especially on large files. AS took me about 25 minutes.
 * 13) On your home screen, your game will now be in a present. Go ahead and launch it once to set it up and create a save file, which we will overwrite with our backed up one from before.
 * 14) Once you've created a save, exit the game. Go into the Homebrew Launcher, from emuNAND, and launch svdt. This time, we are going to select our .cia backup, and to make this easier I recommend ejecting your cartridge, since you won't be able to tell the difference in homebrew. In svdt, the bottom screen will prompt you if you want to enable automatic secure value rewriting. Hit the "a" button to accept this, and then navigate to the svdt folder, and then into the folder which matches the game title. There should be two folders in here, each should have the date, and then a string of six numbers. Whichever folder has the lower six digit value is the older one, and should be the one which you made in step 2. Navigate into this folder, and for each of it's contents, hit the "y" button to restore this backup. Depending on the game, you may have more files or less files than others, just restore any file in white. For example, AS only has the "main" file which I would restore. Hit the "select" button to confirm the overwrite.
 * 15) Boot back into emuNAND and launch your backup again. This time, your old save data should be there!