Really often. Really, really often.
As I quoted in my article on American Amber Ale on the HBT Wiki:
"Add some 80L crystal to Sierra Nevada Pale Ale and you not only have a different beer, but a different style
of beer." [emphasis mine]
One of the easiest ways to change something is to change the yeast, as you've noted. WY3787 instead of WY1056 will make a drastic change! In your case, changing to Trappist or Ardennes yeast will make your Brown Ale an Oud Bruin (well, Dubbel).
I brewed an Enkel Bier (Belgian-style Single) once at a brewpub for which I was brewing. It was their house light beer (Pils with a touch of Vienna, hopped with Saazer), fermented with leftover 3787. That's it! That's the only change. People thought I was a Gawd for developing this light, crisp but definitely Belgian ale. I beamed beneficently and thanked them.
I'd rather bend an established recipe into something different - substituting ingredients, mainly - than try to design a completely new recipe.