I am not related to this family but, this is info gathered from census records.
1910 SD census shows he's age 64, born Australia, divorced, & both parents from Ireland.
1905 SD State census, he's 60, b. Australia, single, arrived US 1859, parents Ireland, been in SD 20years
1900 SD census he's 54, born Dec. 1845New South Wales & parents from Ireland. His marital statis is unclear.
1890 Vet. Schedule Pennington, SD, 57th MA Inf.
1880 Sweetwater, WY has Michael Dwyer 44, b. Australia, wf. Minnie 27 b.IL, & 3 sons, John 5, Joseph 4, & Daniel 11mth.
1870 Albany Ter., WY, Union Pacific RR, has Michael Dwire 24 b. Australia.
Both sons (if indeed these are your Michael's sons) were both in Fremont county WY in 1900 & father listed as from Australia.
NY Ship Passenger records has the ship Chancelor arrived July 1860 with Dwyer,Michael 45, Mary 45, & Michael 17, all from Ireland.
If this is the Michael you are looking for, I believe he married in WY bet. 1871-74 but, left is wife & children. If he was in SD since 1885 & his sons were still in WY 1900, he most likely went to SD alone.
I hope all this helps,
p.s. I believe Michael's age on 1880 census is an error & should be 34.