Does yeast keep same characteristics after multiple pitches?

Just curious.

If you continue to pitch a yeast strain cake on different styles of beers will it keep its unique characteristics or mutate into a common strain?

I have a mild fermenting on White labs english ale WLP0002 I can use the cake in two ways.

Continue to pictch on it with brews meant for the strain. Milds to Bitters to Potter to a Stout

OR

Pitch on whatever the next style is I decide to brew.

Have you considered washing the yeast in the event that you go for a very different style of beer for the next brew? I am not sure of how parts of the yeast cake would affect the next brew if it was a very different style. Many brewers have written that they find washing the yeast too time consuming but I prefer to wash the yeast so that I don't get much of the previous brew's trub mixed in with the next session.

I do a lazy man yeast wash. i will dump the trub in sanitized gallon jug. put in fridge while doing the next brew the same day. when ready to pitch i take sanitized turkey baster halfway into seperated trub and use that yeast. i will fill a couple more mason jars and save in fridge and dump rest of trub in my hop garden.

I do a lazy man wash too, but I sanitize with a steam cleaner, wash with filtered tap water, and just let it sit till it separates and decant into pint mason jars ^__^

Course, if you get off a large batch of washed yeast ( I can get 2-4 pints sometimes) then you can use the yeast indefinably.

I personalty Like my brithish ale yeast more now that its in its 2-3ed gen, since its produces a LOT less sulfer in its fermentation and seems to settle a bit faster than normal.
either way Ive heard that if you alcohol levels are low then you can go up to 10th generation before problems occur, and thats assuming you dont stash a earlier generation in a mason jar and keep it nice and cold in the fridge.

We don't wash but we run ours for 8 generations using 1007 and 1968. I personally would be ok stretching it to 9 or 10 but past that I am unsure without washing.

I like to re-use yeast but typically don't wash it. I usually just pitch right on top of the old yeast cake. With English ale yeast you'll get fewer fruity notes. I've always gone from lighter to darker and always go up in gravity. When I do this I like to start out with a bitter because I can then brew whatever I want after. If I were you I'd go from a mild to maybe a red ale or brown ale to export stout or rye IPA then an imperial stout or barley wine. Just because you use English ale yeast doesn't mean you have to stick with only English ingredients. I personally think that English ale yeast and citrus-y american hops play well together.