A bristle dartboard is sensitive to humidity and temperature swings, so pulling it off the cabinet and keeping it flat is more important than how carefully you wrap the cabinet itself. Handle the board like the delicate part and the cabinet like ordinary furniture.
Pro tip: Rotate a well-used bristle board a quarter turn before remounting it at the new place; it evens out wear from the most frequently hit scoring zones.
Watch out: Never store or transport a bristle dartboard leaning on its edge for an extended period; this causes an uneven warp that throws off scoring accuracy.