Yeast mutations are a product of environment. You can apply selective pressure yourself to encourage particular traits, or you can simply re-pitch multiple generations in a very clean environment while varying the fermentation schedule to effect a change. To then store the yeast long term, you would isolate a single colony from a culture plate, grow it up and verify it performs as you like. Then you can store that culture long term on slants or plates, and continually refresh it over the years.
Simplified example: I start with a commercial strain from one of the "banks" (White Labs, Wyeast, ... ). Let's say it's already a very flocculent strain and I want to maintain that, and I want to mutate it to create a more completely fermented beer. I want to select for the strain with the most "staying power" that is still going to floc well. So, I brew a beer, and I harvest yeast only from the very top of the yeast cake in the fermenter, after the beer has started to clear but still close to the end of fermentation. This should represent the yeast that stayed in solution the longest (still fermenting), while also retaining it's flocculent characteristics (it flocculated, and fell to the cake rather than staying in solution). I then isolate a single colony, and repeat.
If I'm lucky, I can thus develop a strain that does even more of what I like, while retaining the flavor profile of the original. If I'm not so lucky, I get the characteristics I was hoping for, but also get some unwanted tag-alongs like off flavors etc.
There are many other variables to consider, but this is the gist of it.
Classically, new strains were developed "by accident". A brewery would repeat the same process over and over again, while maintaining it's own cultures. Over years, that brewery's strain would become uniquely suited to that environment. This is still happening today, although more and more breweries now re-pitch only a few times, and then start over with fresh yeast from a bank to avoid the cost of maintaining their own cultures. In those environments, any mutations are discarded after a few generations.