New Zealand sits astride the junction of two of the earth’s major continental plates, and is geologically young. The effect of this, though, is the opposite of what has been suggested.
Rather than slowly sinking into the South Pacific Ocean, the South Island at least is still being pushed upwards by the interaction of those two continental plates, along what is known as “The Alpine Fault” (which slices the island in half on an oblique line running north east-south west). Not exactly rocket science, I recall this from Grade 10 Geography.
The Buller River is not the only issue, either. Similar problems are being experienced all along that section of coastline, well out of the way of the Buller or any other river.
What may or may not be happening in Africa has little bearing on this one.