I have found that Clorox wipes are very effective for cleaning out Esterbrook sections... or any tube that has an opening at each end.

I cut the wipe into strips, roll it or twist it, and then thread it into the section. Then it's just a matter of flossing back and forth.

The reason that they are particularly suited to this task is that they are tough. They do not tear, shred, or otherwise fall apart under this major stress.