Is there a definitive method for cleaning headliner smudges?

[Jefsla]

I have a few dirt smudges on the cloth headliner area near the b pillar. Is there any specific formula/solution/ and technique I should be using to clean this correctly? The last thing I want to do is make it worse. Searched, but there seems to be a variety of suggestions. I’m looking for the best suggestion.

 

[Puddles]

I found a few smudges after having the windshield replaced and they wiped off with a damp cloth. I was pretty sure it was just dirt though. If you think it’s greasy, you might have a detailer look at it.

 

[XLR82XS]

HI-TECH extra foamy. Used in one of the BEST detail shops in Metro Detroit since 1985 - and used by me since 1999.

https://www.amazon.com/Extra-Foamy-Automotive-Interior-Cleaner/dp/B01KYFUKUA