Is it possible to fill the joints of a dry stone wall?

Hello everyone,

I have a dry stone wall made of large stones in front of my parking spaces. I would now like to pave my parking spaces and, so that I don't always have dirt from the joints of the wall on the pavement, I would like to fill the joints! Normally, you don't do that with a dry stone wall, right? Is there still a way to do it and what could I do? Attached are pictures of how it looks for me and how I would like it!

Thank you very much for the help!!!

Grouting should theoretically work. Would proceed as with the renovation of an old rubble stone wall

It won't look good visually. It's neither fish nor fowl