The National School system was based upon the Model Training School which trained teachers with government aid at Kildare Place in Dublin. Founded in 1811, this Society for Promoting the Education of the Poor in Ireland would become the Kildare Place Society. The goals of this society were to mix together children of all religious denominations, and after much hard work, eventually achieved its goal. They would organize 1,621 schools for 137,369 students. The national system took over in 1831.
Each local National School kept its own records. Most records for the six counties of Northern Ireland are housed at the Public Record Office of Northern Ireland. The archive has an online introduction to their collections “National Education Records”: www.proni.gov.uk/local_history_series_-_05_-_national_education_records.pdf The National Archives of Ireland is the official repository for education records in the Republic of Ireland. However, their holdings are by no means complete. Their online “Guide to Sources on National Education”: www.nationalarchives.ie/research/research-guides-and-articles/guide-to-sources-on-national-education does inventory the holdings. In the Republic, registers may be kept locally, with the church, at a university or library. Much National School material has been microfilmed and is at the Family History Library (FHL): www.familysearch.org
National School records contain different forms. Usually the name of the pupil, entrance date, age, religion, residence, occupation of the parent, school progress, and withdrawal information. The withdrawal section provides immigration information.
This is a resource worth studying as they contain much information about the pupil. Always check the FHL microfilm collections first to see what can be easily obtained. If not there, then pursue what is on deposit in Ireland or Northern Ireland.