A series of laws were passed in Ireland which sought to “persuade” those who were not Church of Ireland (COI) to change denomination (in the hope this would increase loyalty to the Crown). Known by the collective term of Penal Laws they started to come into force in the mid 1600s. They restricted or prohibited folk who were not COI from all sorts of things such as owning land outright, having a lease for more than 30 years, being an MP, holding public office, having more than one horse, being trained as a Catholic Priest, attending University in Ireland, and so on. They were focussed on Catholics but also affected Presbyterians. They were not routinely enforced, and seem to have been used occasionally to make an example of folk, and were gradually repealed, with the last one - the prohibition on being an MP - being repealed in the 1820s. Nevertheless they were very repressive laws and limited job and social opportunities in Ireland for those who were not COI.
Google Penal Laws for more detail.
The switch between Presbyterian and COI may be more to do with marriage. Tradition was to marry in the bride’s church after which she’d attend her husband’s. So, if a COI woman married a Presbyterian man, the marriage would be in the COI but their children would normally be Presbyterian. And then some people just changed denomination because of religious or personal factors (eg they didn’t like the Minister).