How do you reroll when upgrading tier?

When you go from Legendary to Mythic or Mythic to ancestral. Do you reroll the whole piece cheaply until you get a perk of the next tier, and lock in from there?

Or do you single out the least desirable perk until you get a desirable one at a higher tier and repeat for each single perk, thus ensuring you keep at least your previous roll in any case?