Basically, it depends on the type of polyamorous type relationship you are in.
If it's a relationship where person A is dating person B and C, but B and C aren't dating (this is typically called a V), then it's very similar to how a monogamous breakup goes except person A will still be dating one of the people.
If it's a throuple relationship where all 3 people are dating each other, it might become a relationship like the one mentioned at the start of my first example with A dating B and C but B and C are no longer dating each other.
It's also possible that they might be broken up with by multiple partners if they are in a closed polyamorous style of relationship where they aren't allowed to date someone outside their group and have a rule where everyone must date each other… but this form of polyamory is a lot more rare.