Hello, I don't have the option to migrate at all. I checked the games and neither of them are fake. I got Mewtwo on firered a while ago on my DS-lite, and I reached Pal Park in Heart Gold. Still no "Migrate" option. No idea why not.
First, you have to have a DS (not a 3DS or 2DS) and you must put your Gen 4 game in your normal slot and your Gen 3 game in your Game Pak slot. At the title screen (where it says "Continue Game " and the like) there will be an option to migrate. Follow the instructions (eventually you will have migrated 6 Pokemon), and your Pokemon should be at the Pal Park where you collect them.
If you have done all of these steps then you should have migrated 6 Pokemon. If for some reason it still doesn't work, it may be a glitch.
Hope I helped you in any way.