Suppose that I'm talking about which part of the world someone's family originally comes from.
I believe that "My family is from Ireland originally" would be

Tá mo mhuintir as Éirinn o dhúcas.

But how would I ask somebody else where their family is from? My best guess is:

Cá bhfuil do mhuintir as o dhúcas?

Feels likes something is missing there though. As always, any corrections or alternate suggestions are very much appreciate.

An h is missing and a fada: ó dhúchas

And for orign, the copula is used:

Is as Éirinn (do) mo mhuintir ó dhúchas.
Cárb as (do) do mhuintir ó dhúchas?

OK, now I vaguely recall that family origins is a copular construction in Irish.

And I suppose that cárb is the equivaltent of cá bhfuil in this type of copular structure.

But looking on Teanglann, the examples given for cárb all involve place of origination.
Can you give examples of using cárb that do NOT involve place of origination?

Ní thuigim. Why should it involve something else?

I just find it odd that the word cárb exists solely for this one specific purpose.
Anyway, thanks for your input.

It is actually not a word, just a form of the word cá;
or more correctly the combination of different words (cá + indir. relative form of copula, literally: cá + a + ro + ba > cá + arb > cárb).
Such combinations doesn't need to exist, they are built if necessary.

One could imagine usages outside of "cárb as" sentences but they are very rare so that it is difficult to find examples.
E.g.
Cárb áil leat ...? = Where do you wish ...?
There's at least conditional mood Cárbh áil leat go n‑ullmhóimis í? = "Where wilt thou that we prepare?" in the Bible (Lúcas 22.9). (Peadar and Eoin and the other disciples had perfect Irish )

Or perhaps even:
Cárb é an sagart é? = Where's he the priest?
But this example is totally made up.
You'd probably rather say Cá bhfuil sé ina shagart?

So, Cárb is "actually"cá-aroba. I'm glad it was shortened, buíochas le Dia.
But seeing it "stretched" out like that, it makes more sense to me.

Go raibh míle maith agat.