Drawing the edge through cork, wood, or felt only works on thin "foil" type burrs and it may not leave the best edge behind.
You usually can move on to a higher grit stone without removing a burr from the coarser stone, assuming you want a more refined edge, and I find this the easiest form of burr reduction. Keeping a low-grit aggressive toothy edge (while getting rid of the burr) is more difficult. You still have to get rid of the burr at some point however, and while I think fine/thin burrs are easier to remove they are also harder to see and feel which can make it harder to check your work.
If you create a big burr on a very coarse stone and take that edge directly a finer stone that is on the soft side you can scratch the soft stone, so you would want to reduce the burr on the coarse stone first.
Speaking of soft stones, "mud" can prevent the formation of significant burrs because with edge-leading strokes the apex is plowing through an abrasive slurry. This makes it hard to get a really sharp edge from that stone, but it can set up well for quickly finishing on something like a ceramic rod, without having to fight a burr.
If you have steel that makes a tenacious burr you may want to avoid making burrs as much as possible, especially on the coarsest stone. This can be tricky as if you move to a finer stone too soon you make more work for yourself instead of less, and you'll be tempted to raise the angle because it's taking too long to form the edge.
