You can definitely move up a class.
There are different kinds of working class and middle class which is where the problems with definition come from.
You get the traditional working class, tradesmen, cleaners, shop staff etc. May own a modest house or rent. Not particularly well educated but not opposed to education and often keen for their children to be educated. Normal, just getting on with it people. Most of my family are like this.
Then you get the new money working class. They have made it, often through their own business. Still go to the match, watch sky, do their own DIY and expect their children to work hard too.
Unfortunately you also get people who class themselves as working class but don't work and don't want to. Not even going to go there.
Middle classes are basically decided into two- hippies and normals. I don't think this is very controversial so not going to describe them.
Nothing wrong whatsoever with belonging to any of these tribes, particularly if you're born there. Stuff to be proud of and less proud off in all of these different cultures.
Be boring if we we're all the same.