My experience is that cosmoline gets anywhere that it can, and then some. I completely field stripped and cleaned my SKS for a few hours when I got it, and it still sweats cosmoline during each session at the range. (Particularly from the wood around the handguard and receiver.) If you're trying to get the cosmoline off, I'd recommend a soak in mineral spirits, or a spraybottle filled with mineral spirits to wash it away.
Pretty cool that yours came with a cleaning kit. Mine was missing the kit and the cleaning kit spring in the stock. I'm considering getting a kit and spring just to make mine complete, but I find that a bore snake and a basic rifle cleaning kit are way easier to use. I also don't want any unnecessary wear on my barrel, since my SKS is ancient.