Pokemon games don't have true backwards compatibility, but there is a sort of forwards compatibility. Through various methods, you can transfer Pokemon from generation three games to generation six games!
For those that don't know, Pokemon games are split up into "Generations." There are currently six generations:
  • Generation 1: Pokemon Red, Blue, and Yellow
  • Generation 2: Pokemon Gold, Silver, and Crystal
  • Generation 3: Pokemon Ruby, Sapphire, Emerald, Fire Red, and Leaf Green
  • Generation 4: Pokemon Diamond, Pearl, Platinum, Heart Gold, and Soul Silver
  • Generation 5: Pokemon Black, White, Black 2, and White 2
  • Generation 6: Pokemon X, Y, Omega Ruby, and Alpha Sapphire
Generations are used to show the compatibility between games. Games in the same generation can easily trade back and forth (Although there are Compatibility Issues with X and Y). Games in different generations can only transfer one way to the next generation. At this time, Pokemon can be transferred one way all the way back from Generation three.
Transfer From Generation 3 to Generation 4

Pokemon from Generation three games can be transferred to Generation four games if you have a DS with both the DS and GBA slots. Have both games in their proper slots, and use the Pal Park in Generation four games to transfer Pokemon over.
Only six Pokemon can be transferred over in a 24 hour period. Pokemon with Hidden Machines (HMs) or held items cannot be transferred over. Once the Pokemon are sent, you must find them in a Safari Zone type area. Find them and capture them using a Poke Ball with a 100% capture rate.
Transfer From Generation 4 to Generation 5

Pokemon from Generation four games can be transferred to Generation five games if you have two DS, two 3DS, or a DS and 3DS system. Start up the system with the generation four game, but do not start up the game itself. Start up the system with the generation five game, and start up the game. Travel to the Poke Transfer Lab in the Generation five game and speak with the scientist in the back of the room. The scientist will walk you through the steps you need to take to connect the two systems.
Unlike the Pal Park, the Poke Transfer Lab does not have a limit to Pokemon that can be transferred in a 24 hour period. However, Pokemon with Hidden Machines (HMs) or held items cannot be transferred over. Once the Pokemon are sent over, you must capture them using a bow that fires Poke Balls. The Pokemon will jump from bush to bush in a 2D mini-game. They cannot be captured while hiding in a bush.
Transfer From Generation 5 to Generation 6

Pokemon from Generation five games can be transferred to Generation six games using the Poke Transporter and Pokemon Bank 3DS Apps. The Pokemon Bank app requires a $5 annual fee, and the Poke Transporter cannot be used without it. Follow the steps below to transfer Pokemon:
Note: You can select one Pokemon at a time or click the SELECT button to turn the selection arrow green. This will allow you to grab multiple Pokemon at once. It works in Pokemon Bank, Generation Six games, and Pokemon Black 2 and White 2, but may not work in Pokemon Black and White.
  1. In your copy of Pokemon Black, White, Black 2, or White 2, move your Pokemon to the first PC storage box and save the game.
  2. Open up the Poke Transporter app and transfer all Pokemon in the first box over to the Pokemon Bank's transfer box.
  3. Open up the Pokemon Bank app and transfer the Pokemon from the transfer box to one of the other boxes.
  4. Empty the transfer box of all Pokemon before you repeat the process from step one.
Unlike the Pal Park and Poke Transfer Lab, the Poke Transporter and Pokemon Bank do not have limits to Pokemon that can be transferred in a 24 hour period, and Pokemon with Hidden Machines (HMs) can be transferred over! However, any items a Pokemon is holding will automatically be sent to your bag.
How To Transfer Pokemon From 3rd Gen to 6th Gen - Pokemon Omega Ruby and Alpha Sapphire Wiki Guide - IGN