Got all platforms and unfortunately not. Nothing can transfer.
Though something I reckon they should do is transfer level and all unlocks ONCE. Only once ever, so then if you do change platforms you can have all your hard work but you won't be level 1 so you can't farm the boxes fast. Leave MMR/SR ect on the old platform so you can get the one you deserve. Won't happen though because they'd have to contract with the other platforms :/