If you are looking for someone who was a child in NY from 1854 to 1929 but then seemed to disappear, that person might have been relocated to the US Midwest via a baby train, or what came later to be known as an Orphan Train.
Records were seldom kept, but there are organizations who can help in your search. A list of them are below.
In 1853, a young minister named Charles Loring Brace founded the Children’s Aid Society to help vagrant boys by providing room and board and education. This was a first step toward modern foster care. The society become overwhelmed and as a solution, decided to send groups of children to rural areas for adoption and to help with farm labor.
After send children to farms in nearby Connecticut, Pennsylvania and rural New York, the Children’s Aid Society mounted its first large-scale expedition to the Midwest in September 1854 and continued to run until 1929.