Summary
Hold one of your Assist Buttons to tag out your Point Character and bring in the Assist Character represented by that Assist Button.
Tagging In and Out
This is the most basic and fundamental method for tagging your characters in and out of combat. However, it is probably also the most dangerous method.
Again, Assist Character 1 is tied to the Assist 1 Button and Assist Character 2 is tied to the Assist 2 Button. Hold the respective button to have your Point Character tag out and your Assist Character tag in. When coming into the screen, the Assist Character will come in from the side at about jump height and travel very quickly about half way across the screen diagonally down. Up until they land, they are actually attacking the whole way down. However, once they land, they perform a quick taunt before they are fully active. It's this taunt that makes Cross-Over Attacks so dangerous: if the opponent blocks the Cross-Over Attack or if the Cross-Over Attack whiffs, they are very vulnerable to attack and tagging in your character can mean taking a huge amount of damage.
Strategy Corner
Many times the reason a player may desperately swap out character is because they really want to save the life of another character who is about to die or because they want to have their Assist maintain a power up such as Arthur's Gold Armor or Zero's Sougenmu. So if you block a Cross-Over Attack, your first instinct may be to just land the biggest combo you can think of. But sometimes you may really want to consider another option:
Combo into a Snap Back and bring back the character who just left. If the opponent was desperate enough to try a Cross-Over Attack, they really wanted that character out of there. So ruin all of their plans by bring them back. Thus, their power up can wear off, making them waste a Level of Hyper Meter. Or all the red life the character built up in battle will instantly be lost. And if they were close to death, well, now you can finish them off. Leaving them alive to regenerate all their Red Health can easily come back and bite you. Don't give them that chance. Sometimes it's not about just trying to do the most damage at the time possible.
However if you manage to get the Cross-Over Attack to connect, the hit will pop the enemy into the air with a Soft Knock-Down. Normally, after hitting the opponent, the little taunt you perform will cause you not to be able to combo the falling opponent. However, if your Cross-Over Attack hits at the tip of its attack right before your new Point Character lands or if you hit the opponent high enough out of the air, you can actually get some hits off on the falling opponent before they hit the ground and go for a big combo afterwards. So while there are huge risks involved in performing a Cross-Over Attack, sometimes it can lead to a decent amount of damage.
Also, it is worth noting that Cross-Over Attacks hit high: they must be blocked while standing! If the opponent is crouch blocking, Cross-Over Attacks will hit them so sometimes you can get away with Cross-Over Attacks by pulling them out in unexpected situations. The ensuing attack from the Cross-Over Attack is very quick, so it's not easy to react to random Cross-Over Attacks when done in odd places.
Holding the Button
Because you have to hold the Assist Button down for a brief period of time, it can make trying to swap characters out in the heat of the moment very awkward. Fortunately, Marvel Vs. Capcom 3 tries to makeup for this by buffering the Cross-Over Assist code. If the game detects that you have been holding the Assist Button down for a period of time, as long as you keep holding the button down, it will perform the Cross-Over Assist at the first possible moment available. For example, if you have a Point Character who has only pixels of life left and your opponent activates a Hyper Combo and you want to swap characters so the Point Character escapes and the Assist Character comes in and takes the damage instead, you can hold the Assist Button down during the Hyper Combo activation freeze and, as soon as the Hyper Combo starts, you'll swap characters out immediately.