You could test out a cleaning approach on her legs under her skirt before trying it on something as visible as her face. But that only tests the fabric, not whatever paint or color was used for her features.

I would stay away from water. You might wind up with rings. You could try a gum eraser - they're usually found with art supplies and they're non-abrasive.

Gum eraser might be a good place to start to remove surface dirt. If you do use water, use distilled water only on a clean, soft rag. When we had a microfiber chair, the secretary to the president of the company we bought it from told me that and said it is the minerals in tap water that will cause rings (as well as residual staining).

Good advice about minerals in tap water. One website suggested Magic Eraser. Not sure about that!

Magic Eraser is abrasive -- micro abrasive but still abrasive. I thought they were the greatest first time I used one, but it seems they have become overused and used where they're not appropriate. I've seen people say to use them on patent leather finishes -- and that's a great way to dull or fog the shiny finish. I've never used one on fabric that I can think of, and this doesn't make me think it would be a good time to start, If you do, make sure it is wet and that you tread lightly.

This might be bad advice but, I wash the dogs cloth toys in a king size heavy cotton pillow case with a heavy duty rubber band on the end to keep it shut. I use a gentle no fragrance laundry soap on the gentle cycle. I also dry it all in the same pillow case on low heat. The toys come out nice and new.

FWIW, I wasn't thinking it's the minerals in tapwater that might leave rings on the fabric - it's other surface soil, and possibly pigment as well.

You could also just wipe/brush the surface carefully with a microfiber cloth, before trying anything else.

Since muslin is just a cotton material (plain weave), you could possibly practice on an old piece of cotton cloth first to see what is the safest choice. Agree, no tap water on the old doll.

In the photo that's linked, the entire face looks like it's pigmented...?

I wouldn't use any kind of water at all. Wetting the fabric may cause discoloration regardless of what kind of water you use. You have to consider the soiling and body oils that are already present in the fabric.

You may want to try one of this type of erasers and use gently.

https://smile.amazon.com/Alvin-1248-Professional-Drafting-Cleaning/dp/B004O4I3EI/ref=sr_1_11?keywords=Gum+Eraser+Cleaning+Pad&qid=1653244743&sr=8-11

I have one of those types of erasers - it's basically a bag of eraser crumbs and you squeeze them out onto the surface and rub them around to remove smudges, then brush off. Mine is pretty old but still works - it looks like this:

I thought about suggesting it, but wasn't sure if it was non-abrasive.

Lascatx, I use them but definitely not on all surfaces. What I I've noticed is the original Mr. Clean's Magic Eraser isn't as strong as it used to be. And don't even bother with off brands.

Carol, do you mean the paint on her face that's pigmented? The doll in the link looks too clean and maybe never out of the box it came in. My doll is a bit faded where she has paint. I wouldn't dare try to clean the paint.

I'm going to try a slightly damp washcloth and if it doesn't help I'll order the dry cleaning pad.

That looks like some celebrities on their social media accounts vs. seen out at the supermarket.

Lol, Lars and pal !

Who, Khloe? One of the Pod People Sisters?

Lars, reminds me of that NYC (?) woman who could not stop getting facial cosmetic surgery. (Her name id on the tip of my (natural) tongue.

You could ask vintage quilt collectors and antique doll collectors/preservationists. Another option is to just leave it.