Neutral mission tokens can be retrieved and dropped by figure carrying one on either stockpile, they do not disappear.
So, during the mission any figure (except ones with the Non-Sentient ability) can retrieve tokens from either stockpile. I.e. your stockpile will probably need some guarding if you intend to keep yours.
Note that because the retrieve and drop are performed without requiring actions, a figure can share a space with another figure to perform them as long as they have enough movement points to get into a valid space to end movement afterwards.
Why it cannot be Non-Sentient figure ? That figure cannot perform "use" action. During this mission you don't make "use" action like in case with doors, crafts or terminals, you just pick up for your movement points. So Nexu also can take barrel in my opinion.
Jabba's Story Mission 1: Hostile Negotiations.
in Imperial Assault Campaign
Posted
Why it cannot be Non-Sentient figure ? That figure cannot perform "use" action. During this mission you don't make "use" action like in case with doors, crafts or terminals, you just pick up for your movement points. So Nexu also can take barrel in my opinion.