I got married two years ago. It was a civil ceremony. To us, it meant that we wanted to support each other emotionally and financially and have the legal means to.
If I have a ceremony with another partner, here is how I assume it would go:
My husband would walk me down to the partner I am "linking myself" to. We would say our vows to each other, to be committed to be there for each other emotionally and financially.
But on top of the "I do" we would say to each other, all three of us would have an additional "I do". My partners would both say "I do" to some text about recognizing each other as equally important to me and having a relationship with each other through me.
I would have to say "I do" to a text saying that a view them as equal, important individuals, and agree not to use one of them to hurt the other.
This is off the top of my head, and we would obviously work on it more, but I think each new person entering the relationship in an "official" way would require all existing members to take part in the ceremony in some way. Hopefully I am making sense.