Thanks for your post to the Ireland Reaching Out message board.
The parish register for Carron is held within the Diocese of Galway. It is available online for free under the National Library of Ireland athough, it only begins in 1853. Here is the link: https://registers.nli.ie/parishes/0617
I see that the baptism register has been transcribed through the Clare Library website and here is the link:
You may already be aware of this. You might look to see where this family resided before they left for the U.S. and/or Australia. If there were other family members in the same area, they may have remained. You can then trace them forward through the 1901 and 1911 Censuses.
Please let me know if you have any questions and I will try to assist you. If you have not looked at the Clare Library website, you should definitely take a look at it.
The very best of luck with your research,