Do the mechanisms for mitosis and meiosis evolve or have they remained unchanged for millions of years? If the latter why?

mitosis and meiosis are both incredibly well conserved in almost all higher organisms studied. given the degree of evolutionary time that has passed to optimise these processes there will be few if any pressures for change. most if not all mutations will tend to make the processes less efficient and this will be selected against.

That said, it is possible that one or more new mutations might arise at some point in the future that could make the processes more efficient.