Hello Shay,
It looks like you've traced back your cousin's ancestors for quite a bit already. Well done!
Are you trying to find information on Christopher Dunleavy's family, both further back in time and for generations that followed?
For records before 1864, it would be the relevant genealogical centre that should have the records you're looking for, if they still exist. Due to the high levels of emigration from Co. Mayo, it has two genealogy centres: The North Mayo Heritage Centre and The South Mayo Family Research Centre. Because of this, it's great that you know where Christopher Dunleavy was born. The relevant genealogy centre would be the
North Mayo Heritage Centre.
You can get in touch with the heritage centre to request further help with your research. You can find this and other useful information in the
Starting your Genealogy journey thread as well.
There might be someone on the Community Forum who can help you out further, but I do hope this points you in the right direction and helps with your research.
Have a great day!
Noortje