Maybe this is a frame challenge:
It's possible that the people don't care about continents as such at all. They probably are more interested in what happens on the surface: where they build their homes and cities, where they can grow food, hunt, fish, where they get resources (wood, coal, metals etc.), and how they can interact with each other, be it friendly visits or trade, or defending against rival or hostile civilizations.
Whatever happens underneath is of importance only if it causes something on the surface, eg. volcanoes erupt, earthquakes destroy their houses, and tsunamis drown their coastal cities. And, even then, plate tectonics and such could be secondary, unless the volcanoes and earthquakes are so common that the people have to understand why they happen and how to predict them.
That said, the borders probably evolve over time as tribes work together, wage wars and form alliances; and alliances become nations, and nations work together, wage wars and form even bigger alliances and so on. Geographical factors drive this evolution as trade routes bring them together and competition about resources put them against each other. Distances, practicality and such factors probably matter more than anything else.
When there are many islands, sailing and navies and navigation play a big role. Those who can build better ships fare better, as do those who can get their ships intact from port to port. Nonetheless, this is all evolution, and its course cannot be predicted, so in the end, it's up to you to figure out how it has affected the nations and their borders.