I can see two possibilities:
- The minion's deathrattle is replaced
- The minion is given an additional deathrattle
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From the Hearthstone Wiki: Minions may benefit from multiple Deathrattles at the same time.
In your example, Cairne's bloodrattle would go off first, summoning the 4/5 Baine. After that, Ancestral Spirit would resummon Cairne.