I couldn't miss the chance to practice my drawing!
Transmutation is a feature in Final Fantasy Airborne Brigade. One special property of event equipment is that they can be transmuted into a higher-rarity one. The lowest they can be is R, which can be turned into an R+ version of the same item. R+ then turns into a different, SR item. Finally, the SR can turn into SR+.
Transmutation is done using evolve items, and the amount consumed will vary by equipment. In the Final Fantasy XIII-2 event, the amounts were static, meaning that if it costs 200 fragments to transmute one item from SR to SR+, it will cost 200 no matter what you do to that item. Starting from the Final Fantasy VI event, however, the more the player limit boost their equipment, the lower the cost will be. The less rare abilities and weapons cost 200 evolve items unboosted, and lower to 100, 50, 25, and 10, for each additional boost. The rarer abilities and weapons of each event, however, cost 400 evolve items, and lower to 200, 100, 50, and 25. The costs are usually half of that for R-rarity items going into R+.
Event weapons usually have stat values of around 10,000 atk/def at level 200 SR for less rare ones, and 15,000 for the rarer ones; and around 15,000 atk/def at level 240 SR+ for less rare ones, around 20,000 for rarer ones.
As a note, during the main parts of most events, there is an enhancing increase campaign, where enhancing weapons and abilities will give 1.5x enhancing exp.
Transmutation means to change from one form, nature, substance, or state into another.