I haven’t yet found any links to Australia, but I’ve only been concentrating on direct lines thus far.
I’ve seen Ballymote come up a number of times in my research, but can’t say where at the moment. I just took a quick look at my tree and I do have a Patrick James Davey born 1844 about a stone’s throw away (~2 miles)in Carrickbanagher, Sligo, Ierland. Father was Charles from Limerick.
I'll take a deeper look this weekend to see if i can't find more.