Seems to me caiafa has it.
That is the way we do it in bread baking. Freeze cultures that have the characteristics we want and if the current mother picks up characteristics we don't want we toss it and revive from the frozen line. A yeast culture will change over time but if you use it for the same purpose, using the same ingredients every time, it shouldn't change much past a certain point unless it gets infected with something.
If you want to keep a yeast strain at a relatively stable state, you need to feed it regularly and watch its activity cycles. The moment fermentation slows (not stops) you need to add more food, because at that point the other things in the culture start to gain ground. If you want to slow the growth of other things you have to refrigerate it.
There's also yeast washing
, which seems like a more complicated and probably more certain way to maintain the exact strain you want.