This is not yet confirmed but I am assuming any items earned through the BattleNet account link will pass over. Note I refer to any items only earned that are sent to all three platforms including OWL Tokens, Heroes of the Storm Nexus Challenge Rewards (which all of which are now found in standard loot boxes), or Twitch Streamer rewards like the Baptiste Reunion Challenge sprays.
Please keep an eye on this master topic where I am tracking questions and known facts about the Switch release of the game: