How do you keep a brick walkway from sinking/being overtaken?

Not sure what you mean. What is under the walkway?

If it's laid and tamped properly, it really shouldn't sink. It might be that whoever put them there, may have simply dug them into the dirt. That will make them sink pretty much every time.

Install it correctly. :) If it is already sunk and you want to repair it- you can pull up the brick- and then reinstall it.

Yes, and be sure the drainage is adequate, too.

I find that my stone planting circles sink also. I set about to restack the center circle and found the stones had actually sunk several inches. I am no expert (obviously) but I think stacked/laid stones sink if on a non-rock base. I should have poured a concrete foundation on which to lay the stones. We have a concrete walkway that was not in good shape which we overlayed with pavers 25 years ago and they are still in perfect shape. I believe it is because of the concrete base under them. You may want to take them up, pour a base and relay them.

They don't generally pour concrete under pavers/bricks. Rather they used a crushed stone and tamp it down and it gets as hard as concrete basically- but without the expense mess and forming requirements of concrete. Plus it has a little natural give which helps keep pavers from heaving and cracking- and it makes repair possible over time.