I have a really stupid question, do coral reefs migrate north and south?

So what sprung this question to mind was remembering in a book about trees and forests, that as the climate changed, the entire forest would slowly migrate north or south chasing the optimal conditions. As the climate heated up, pine forests would move north to stay in the colder weather, and if the climate cooled, then tropical rainforests would retreat south to the equator.

*(and the inverse being true in the southern hemisphere.)

**(and yeah obviously not the trees uprooting and walking like Huorns, but rather the saplings growing on the advantageous side of the forest would thrive whilst the saplings on the disadvantageous side would wither, so over tens of thousands of years the forest "migrates" in a direction)

So i was wondering whether this is reflected in coral reefs? Do we see in the world today the reefs further away from the equator surviving the effects of climate change better whilst the ones closer to the equator suffering more? Are reefs starting to grow in areas we originally didn't observe them growing in? (i'm guessing this one would be hard to know though seeing how young the field is).

Could gigantic reefs such as the great barrier reef survive by moving slowly south to the antarctic? Could we as humans, artificially move reefs to new areas that we've identified they potentially could survive in, if for instance there isn't a path for the reef to migrate to naturally. ( in this example, would be like a pine forest getting trapped against a river too wide for it to get across, but if it did the forest would survive on the other side of the river so we artificially carry a diverse range of pine cones and plant them on the other side, helping the forest in its migration)

There was no flair for "questions from the philistine public" so i picked research