It's been a while since you posted so I hope this reaches you. I think we may have some relatives in common/be related.
I have been researching my grandfather's family and just found some Dunkley's. They are Ernest Dunkley married to Laura Maud Sanders. The children listed with them on the 1901 census are William, Harry and John. I found them as immigrating to the United States in April 1920. The children on the passenger list with them were Harry and Rose Ellen. Their final destination was listed as Ann Arbor, Ypsilanti Michigan to a Mr. Sanders (father-in-law/parents). Their nearest relative was listed as Mr. J. Dunkley, son, Birmingham.
Laura Maud Sanders was the daughter of John Sanders and Hannah Poynton. She is the sister of my great-great grandmother Florence Hannah Sanders. Florence was married to Albert William Rayner and they immigrated to the US in 1905. They lived in Pontiac, Oakland County, Michigan.
I would love to exchange information if this fits with your family.