> I presume it picks up local yeast as it grows and eventually becomes a "local" variant

I had a funny experience of exactly this recently.

I got a starter from a friend a year ago and have been feeding it and using it regularly since. But recently, my father had some health issues and I had to go live with my parents for a couple weeks to help out, back in the small rural town they live in. I took my starter with me, and made a couple loafs of bread for my parents, feeding while I was living with them.

But when I got home, my starter was different. I grows way faster now. It's much more active. It seems to generate a lot more CO2 than before.

I suspect during my visit, my starter picked up a new strain of yeast which has joined the ecosystem in my jar. And it's made for some great bread.