Once your partner reaches 0 you can attempt to pick them up by tapping the X button while standing beside them. You are given one medical kit at the start of each mission to do this, therefore you can only do one per mission. The AI often forgets to pick up health items and things of this nature, making it easy to do especially early in the game when they have poor gear. If you've defeated the boss and your AI is downed, it still counts if you revive them in the few moments they give you to pick up your loot and this saves you the hassle of worrying about enemies interrupting you.