I would say celly thinners, although depending on your airbrush (rather it's seal material) soaking might be unwise.
I would put some celly thinners in, flush this through the brush, let it stand for a few minutes, flush the remainder out & then remove the needle & check that there isn't any build up on it around where it travels through the needle seal - if there is a few wipes with thinner should remove it.