"Draw polishing" is a bastardization of a filing technique wherein a SINGLE-CUT FILE is held at 90degrees to the workpiece and "drawn" using both hands. If this makes no sense look up "draw knife" in an online tool catalog to see what I'm talking about.
Draw filing is a way to remove the ripples inherent to turned stock (like barrels) using only hand tools.
I warn you that you must in some way examine the barrel for ripples because otherwise you're gonna' spend a MONTH (I'm not exaggerating) polishing this thing only to have every ripple and imperfection LEAP to your eye like a jumping spider....... you'll hate it.
Whoever told you that "draw polishing" will remove ripples is smoking something green or gold
........... Now if you spend a LOT of time long-stroking the barrel with successive grits of backed sandpaper you can remove ripples, over time.... LOTS of time.
Generally speaking, once you've got the barrel lightly polished and examined for smoothness AND YOU'VE ADDRESSED THE RIPPLES, you'll want to polish it using a spinner. You will work the spinner at 30-50degrees from horizontal and you will alternate or "rock" it periodically. You'll spin it out one way and then tip it the other way and spin out all of your marks.
IF there's a secret to proper polishing it's that you constantly move the piece but let all of the grit lines form in one direction. When ALL of the lines are running one way you then turn the workpiece ("rock" it) and polish all of those lines out until ALL of the lines run the other way.
For your absolute final polish it's sometimes advantageous to run parallel to the bore of the barrel ONLY because the scratches that you leave will be harder to see and less intrusive.
Polishing an entire project this way not only WON'T WORK, it's dangerous. Running the piece this way is exhausting work and you'll sooner or later end up with your knuckles or face tangled in the wheel. Buffing wheels WILL leave a mark on your delicate complexion...........
al