They escorted him to a ship, I suppose whoever was on the ship escorted him from east watch by the sea to the wall.
I'm not trying to belabor meaningless details not explicitly explained on the show, I'm just saying the whole dismissal, seemed very unofficial to me. Did Tyrion even, technically, have the power to dismiss him like that? It seemed like something he did in the middle of the night so no one would be around to stop him (I recall Cersei being slightly ticked about it). If I'm Slynt, I'm deserting, I doubt he'd have faced any real consequences. Then, again, he seemed to enjoy sitting on his high horse as a lord at castle black, so...
just seems like such a raw deal (even without the white walkers and free folk attacking) spending the rest of your days with a bunch of society's scum and no women? I would've Jorah-ed my way right up out of that situation
I remember that scene where he got dismissed cause it's one of my favorites. I'm thinking he chose the wall cause it was closest after they dumped him. He figures he's a high ranking officer or whatever that wouldn't need to do shyt cause he spent his time in King's Landing not believing in any of the shyt that was north of the wall.