Storing NBT in a broken block

How can I break a block and have the dropped gem have the same NBT tag? I am trying to make a mod with warming and cooling metals in order to forge them and I made a bloomery, I want the hot bloom produced in there to continuously cool even if broken. I now have three different blocks with respective textures: "hot bloom", "warm bloom" and "cold bloom", and I have a number NBT called "temp", in every update tick, hot bloom gets a +1 on "temp", and when it hits 5, it replaces itself with "warm bloom" and a similar mechanism works with the cold bloom. I want it to continue adding the "+1" when the block is broken and it becomes a gem or it gets picked up by the player and becomes itemstack.