Hey guys, this will be a quick tutorial on how to claim a Mystery Gift on any of your Pokemon games.
First, you have to know the date of an event, of course, and depending on whether it is a Mystery Gift claimed over Wi-Fi, or at a given location, the process will vary.

I will start with the process of claiming a Mystery Gift via Wi-Fi.
First, when the date of the event arises, you can begin the process of claiming your Mystery Gift Pokemon.
You'll have to go to the main screen of your game, and click, 'Mystery Gift'. After clicking the Mystery Gift button, a screen will appear, giving you the options of : Receive Gift, Check the Card Album, Info, and Quit. You'll want to click, 'Receive Gift'.
By doing so, the game will ask you this, 'The communications channel must be opened to receive Mystery Gifts. Is that OK?'. Click, 'Yes'.
A new screen should appear giving you these options : Get Via Wireless, Get Via Nintendo WFC, Get Via Infrared, and Back. You want to click, 'Get Via Nintendo WFC'. It will then ask, 'Connect to Ninendo Wi-Fi Connection?' click, 'Yes'. Your game will then automatically start searching for a gift, giving you the waiting screen, 'Searching for a gift. Please wait...'
If there is a Mystery Gift to be found, a new screen will open, giving you your choice of Mystery Gift. Once you select the Mystery Gift you would like to claim, the game starts claiming the gift. You will be redirected to your Card Album once the Mystery Gift is claimed. You will then, to claim your gift in game, start your save file, and walk into a PokeMart. There will be a man in green/blue [depending on the Generation] standing next to the door. You will contact him [with an empty slot in your Party] and he will deliver you your Mystery Gift.

Next, I will talk to you about how to claim your Mystery Gift via a selected location.
This process is almost exactly the same as claiming a Mystery Gift via wireless, so I won't go into as much detail, but be sure to read the section about claiming your Mystery Gift via Wi-Fi, so you can fully understand this part, as well.
You'll have to go to the main screen of your game, and click, 'Mystery Gift'. After clicking the Mystery Gift button, a screen will appear, giving you the options of : Receive Gift, Check the Card Album, Info, and Quit. You'll want to click, 'Receive Gift'.
By doing so, the game will ask you this, 'The communications channel must be opened to receive Mystery Gifts. Is that OK?'. Click, 'Yes'.
A new screen should appear giving you these options : Get Via Wireless, Get Via Nintendo WFC, Get Via Infrared, and Back. You want to click, 'Get Via Wireless'. It will then say, 'DS Wireless Communications will be launched.' click, 'Yes'. Your game will then automatically start searching for a gift, giving you the waiting screen, 'Searching for a gift. Please wait...'
If there is a Mystery Gift to be found, a new screen will open, giving you your choice of Mystery Gift. Once you select the Mystery Gift you would like to claim, the game starts claiming the gift. You will be redirected to your Card Album once the Mystery Gift is claimed. You will then, to claim your gift in game, start your save file, and walk into a PokeMart. There will be a man in green/blue [depending on the Generation] standing next to the door. You will contact him [with an empty slot in your Party] and he will deliver you your Mystery Gift.



Basically the same for the both of them, just a few minor changes.
Happy receiving, guys!!!

Nintendo WFC Mystery Gift, 'Virtuoso Victini!', was released on the date this was posted.