So I got my 2023 EQS 580 SUV used and there was a vibration at certain speeds. The dealer replaced the tires and the vibration is gone but I'm noticing when I go over potholes or drains (which we have a lot of here in Chicago), that there is the initial feeling of hitting something but then there is a bit of an after bounce. The dealer replaced the tires and did an alignment and the tire pressure was set to 48/43. I corrected that to 41/41 and the "after bounce" is a lot less but I'm still feeling it. The car had summer tires when I got it and now I have the Pirelli Scorpion Zero tires on it which are run-flats I believe. I went back to complain but the mechanic drove the car with me and didn't find anything unusual. At first I thought that the alignment wasn't done properly but maybe it's just a run-flat tire thing that I'm feeling? I had a 2017 GLS 450 with 19 inch wheels for the last 7 years and never felt anything like this.

Have you experimented with the drive modes?
I found that in Comfort mode, the suspension had some post bump bounce or "after-jiggle". In Sports mode, that disappears, at least on my EQE SUV.
If you find this to be the case, you could go for "Individual mode" and make it all behave like Comfort except the suspension, which you could set to Sport. This is how I have set mine up.

Same here. Sport mode suspension setting yields significantly more controlled motions.

Yes I tried different modes and the EQS has air suspension standard, so it drives a lot different than the EQE. I had one as a loaner for a week. The bounce I'm talking about is not the whole car it's just the wheels.

I had a hard time figuring out if the tires are run-flat or not but with the SKU number on the invoice I figured out that they are not run-flats. Maybe I'm just not happy with the tires

It should say RFT or Run-Flats, or MOE.