I don't know the technical language for what Brittain Ashford does in those moments, but I know exactly what you're talking about. To me, it adds a slightly folk or old-time music aspect to her singing, like Joan Baez or Emmylou Harris, or someone like that.
It's idiosyncratic enough that I totally get how some folks wouldn't like it. But I liked it a lot, in the theater and in the recording. But I'm a sucker for an "interesting" vocal sound. I miss all those character-acting-style voices from the bygone era that you don't hear as frequently anymore — folks like Carol Channing, Charles Nelson Reilly, Glynnis Johns, Elaine Stritch. Those are some famous ones, but cast albums used to be full of random folks with fabulously unique voices.
We do still have a few unmistakable voices among us, with long, successful careers — Mandy Patinkin, Harvey Fierstein, Bernadette Peters. Their voices are a huge part of what makes them distinctive. (And if this idea interests you, here's an old thread that discusses the same thing: https://forum.broadwayworld.com/readmessage.php?thread=976305.)
Anyway, I'm glad when new talents like Brittain Ashford come along who aren't afraid to let their different sounds be heard.