Once when I asked a father how long his child had had a particular condition, he told me "depi li fet, depi li tombe a ter". Is "tombe a ter" used generally for an idiom meaning birth, or was I just listening to a really colorful speaker? Mesi!

We do not generally use the term tonbe atè in reference to humans. This father might have been a farmer.
We use the term mete atè (to give birth to animals, such as a cow giving birth).
So people may say something like:
Manman bèf la met atè jodi a.
The cow gave birth today.
and subsequeltly:
Depi bèf la tonbe atè li leve kanpe l mache.
Once the little cow is born it gets up and walks.

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