Question: Emigrant = someone who is leaving their country.

??? = the country from which the emigrant is departing.

I want to say Émigré country but I don't know if that makes sense.

Maybe country of emigration? But that's too wordy.

Is there *one word*?

Answer: 
I would say "[**country of origin**](http://en.wikipedia.org/wiki/Immigration_to_the_United_States)."
In most cases, it would be **homeland** or **motherland**.

But "**[old country](http://www.merriam-webster.com/dictionary/old%20country)**" is used also

> 
> an emigrant's country of origin
> 
> 
> 

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Additionally, **"source country"** is used in immigration related or technical sources