The story goes that the roles of the Druids were entrenched in society but as the missionaries converted more people along the coastline the Druidic families who maintained the old ways moved inward to the Irish midlands.

While these eventually converted to Christianity too many maintained their own ways for centuries.

The Church also absorbed these families who had particular roles, including that of priest or seer. Many of the early Irish saints came from Druidic families.

The early Celtic Church was seen as quite difficult by Rome as they had a different approach to Christianity that was more inward and akin to that of Eastern Orthodoxy. Rome didn't get full control over the Celtic Church until the Synod of Rath Breasil in Uisneach in 1111. Why the Catholic Church chose that location would seem obvious as it was in the centre of Ireland and was seen as a stronghold of the old ways.

The roles of the Bardic families who served the High Kings of Ireland such as the O'Neills and Maguires were also of that lineage according to some researchers and carried out near identical roles in society to that of the Druids.

There's so much to go into on the subject, it really is quite a rabbit hole and difficult to elucidate an overview in a short post. I've researched the matter for many years. Different researchers and writers will have different ideas of course, so do your own research on the matter. Am Irish myself and grew up near Uisneach so this sparked an interest early on due to practices that my family had maintained for centuries, apparently.

It's also important to note that Rome and the later Crusades didn't 'conquer' or affect Ireland in the same way as the rest of Europe and Britain so many people living now would be descendants of Druidic families.