Card Rush
You select 3 of your Dream Eaters - each has it's own skills, which are the same as in normal game.
The goal is to get the HP of all opposing DE to 0.
Flicking the card up makes the DE use the skill, you can flick more than one to get a higher number. Each card flick takes one portion of the pink gauge and if it's empty you can't attack/defend. The DE with a higher number during an attack breaks the enemy's attack. Mind you, some of those skills have small AoE, so it's possible to miss, even if your number is higher. You can flick more than one card for a single attack, but you have to do it quickly and it takes more of the gauge.
Flicking the card down makes your DE defend and doesn't use the gauge. The thing is, all those skills have execution times, so you have to be very cautious of when to use them.
When you run out of cards (you actually have more than those 4 cards on screen), you can hold the DE icon with your stylus to recharge, but it takes a few seconds.
You can switch between your DEs and I think it also depletes one section of the gauge.
When 2 DE attack with the same number, a lottery game begins - it's basically a match 3 icons flicking contest - the first person whose icon makes the 3 upper icons the same wins and deals "huge" damage (the tutorial says it's huge, but uh, not really).
I think the numeric values on the cards are random - there's no "deck" to be modified anywhere, you just select your DEs.
All in all, it's a fun minigame and the only way to play against someone in multiplayer (local only, I guess).