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The tool used to dump games is Wood Dumper.

Transfer that file to your Slot-1 flash card.
Run it.
When it tells you to "Set a target card", take your Slot-1 flash cart out and put the actual game cart in.
Press A, and then look for an IP address on the top screen.
Go to your computer, and open Firefox3+, IE8+ or Chrome 5+.
In the address bar, type "ftp://", then the IP address listed on the DS, followed by a colon, then the number 21. For example something like ftp://192.168.2.4:21.
If in firefox you get a message that the port is blocked, you need to unblock port 21. If not, skip this step.
In the address bar, type "about:config" with no quotes and press enter.
Agree to the warning.
Right-click in the window and choose New -] String.
For the name, type "network.security.ports.banned.override" with no quotes.
For the value, use "21" with no quotes.
Restart firefox and try again.
Just press enter if it asks you for a username and password.
The screen that shows up should list the game as a file. Click it to start downloading it from the DS.

The tool used to dump game saves depends...
For 3DS saves, you will need DSaveManager.

For normal DS saves, the tool is NDS Backup Tool WiFi.
(For Heartgold/Soulsilver/Black/White, you need this tool.)
You will also need SmallFTPd 1.0.3.

Transfer both the program and it's .INI file to your flash cart.
Find your computer's internal/local IP address. In your start/globe menu, go to the "run" command. If you're on vista/7, you'd use the little white box near the bottom. Type in "CMD" and press CTRL+SHIFT+ENTER. In the black box that comes up, type "ipconfig" and press enter. It'll list a series of four numbers as your IP(v4) address, like 192.168.2.3. Write those down.
Edit the .INI file that comes with the backup program. You can edit it with a text editor like notepad. Change the IP address to the one you found in the previous step, and change the port number from 21 to 8080. Change both the user name and password to "test" with no quotes. Save the changes.
You may need to go into your firewall or other security program and allow access on port 8080.
Open SmallFTPd.
Go to Settings - General, and set the port to 8080.
Go to Settings - Advanced, and check "use local IP".
Go to Settings - Users. Choose to add one, and make sure it's username and password are both "test" with no quotes. For "directories", use "c:\temp" (no quotes) as the physical path. Give the user permissions LRW.
Save the settings.
Click the play button, and the server should start.
Run the backup tool on your DS/Lite.
It should connect to your computer. If it does not, double-check your settings and security software.
Take your flash cart out of your DS.
Put the real game in your DS and then press A.
Press B to dump the save.