I found a newspaper article dated June 16 1893 in a relatives papers that said the children died of Scarlet Fever. The name of the newspaper was not noted.
The article said that the Finch children were buried in the Mill Fork Cemetery which is a very small cemetery. I contacted the caretaker of the cemetery and he said that as far as he knew there were no others buried there that had died of Scarlet Fever. The little cemetery contained mostly his Atwood family.
This is the only source I found regarding my family. My only suggestion would be to search local newspapers or contact the local historical society.
I just did some research on a completely different family that lived in New York. Four of their children died in the fall of 1896-winter 1897 and were all buried in the same cemetery. I contacted the local historical society and asked what was the cause of these deaths. She responded that there was a bad influenza epidemic during that winter which caused the deaths of about 100 people in the county - mainly children. All were buried individually - no mass graves.