show an example of "falling apart'"

I think there is a risk of your counters breaking if they are stone.

Unless the boxes themselves are falling apart, you can have them refaced to look like new. And it could be quite difficult to remove the countertops and reinstall them without the potential for breakage, so you should have it in your budget to buy new ones if the contractor can't or won't re-use them.

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Your risk all of the other stuff being broken or unusuble. Plan a gut.

The uppers will be the easiest. So that part shouldn't be the problem. The challenge will be the counter interface with the tile and the lowers. Usually, the tile to counter seam is caulked. You can cut that. The sink can be dropped and the faucet removed. After that it depends on the counter configuration. Was it installed as one piece? Or is there a glue seam? Was it glued down? The good part is the lowers can be damaged to loosen the counter. Installing the new cabinets, hopefully mostly drawer units, will need careful shimming to snug the counter up to the tile. You should go over this with the cabinet manufacturer to select experienced installers. You need guys with counter installation experience.

We gave away our old granite counters when we replaced our kitchen cabinets and were able to pull them off without breaking them. I‘m sure it would be best to have a countertop person do it although we had no problem DIYing them.

I don't know how backspash will fare as they are put in after countertops. It may be best to take off bottom tile.