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We're talking about normal dish soap for hand washing (eg. Dawn).

Dishwasher soap is a whole different story. It contains abrasives, among other things.

Dawn definitely damages the patina over time. You can get away with a few washes, but then food starts to stick, it rusts, etc.

S/he said soaps, not dishawashing soap.

Dish soap is not dishwasher detergent.

Please don't put your pans in the dishwasher, not matter what they're made of.

I put my tri-ply stainless pans in the dishwasher all the time.

Why shouldn't I?

You should, it's 100% fine (and the main reason I prefer stainless to cast iron).

While stainless steel is “dishwasher safe”, dishwasher detergent is still overly harsh, can cause discoloration, and will prematurely wear the pan surface.