There are 4(or so) types of assist spells in game.

Healing Skills: Pretty much healing spells that you can use to heal someone's HP or SP.
Recovery, Holy Shield(angela), Magic Recovery, etc.

Buff Skills: These skills either increase a stat or power of your team member(s) or either block/shield you from enemies' attacks.
Fiery Attack, Hotfire Attack, Stone Wall, Water Shield, SpeedUp, etc.

Debuff Skills: The skills that you cast on an enemy. You can either block them(seal) or decrease their stats/power to weaken them.
Shrink, Freezing Spell, Sleep, Terrafication, Tree Bind, Mess Spell, Cord Spell, Coma Spell, etc.

Anti-debuff Skills: This is how I call these skills. These skills are the ones that remove the debuffs that your enemies cast on someone at your side, or the buffs that your enemies cast on another enemy at their side.
Free From Mess(and others), Unstun, etc.

So in this case, to burst, the person(either pet or player) must have buff or debuff skill(s).