One of the ways the immigrants kept their dreams alive was to tell of the country they left behind. Using Chì Mi Na Mórbheanna as a lullaby is certainly consistent with that aim.

The Jim MacLean fellow is who made a lullaby song out of the tune. Sounds like it was a fairly recent song. Wonder when he wrote it.

Anyway, yes, you have the facts correct. I've looked high and low for the "Duil ri Baile Chaolais fhaicinn". I am not sure if it was just a tune or a song.

If you do come across it, please let us know here.