Laura: In the, abt. 1876 time frame you mention, there was only one or two possible KIRKLEY families in Falls County, Tx. In the 1970 Falls County census there is a J.E. Kirkley who had come from South Carolina. In the 1880 Falls County census, he is still there along with Robert Kirkley and family, also from South Carolina.
The Robert Kirkley is my great grandfather and he and his family arrived in Falls County by wagon from Sumter, South Carolina in about 1877. The other Kirkley family, that of J.E. Kirkley probably arrived there right after the Civil War. I have not yet determined their relationship but they were probably brothers (or at the very least, cousins) My Robert Kirkley and his family moved to Nacogdoches, Nacogdoches County, Texas in about 1884, leaving J.E.Kirkley and family in Falls County.
I know very little about J.E. and his family, but the census lists his children as having been adopted.