To start air combos with assists, you hit the launch button with a direction: UP for more damage, DOWN to accumulate more super meter, or LEFT/RIGHT (balances damage and super gain) and then the assist button. Also, you counter an air assist by hitting the same button your opponent does when attempting one. So, if you're playing someone who uses them a lot be prepared to press one of those 4 directions + the Launcher Button when getting air comboed.
