I haven't actually read MotW, but this question was unanswered so I thought an AW-based answer is better than none. That answer is: probably, but it depends. Physically take a new playbook, and start copying over stuff from your old playbook. If the new playbook tells you "you start with two of these moves", then choose them. Do you keep everything you had before? That depends. Is it part of what you left behind? Is it part of what's the true you, regardless of your previous life/specialty?
Technically, this can apply to the "potential for improvement" represented by checking off improvements. However, I have a hard time thinking up an example of a character who has taken the two "take a move from another playbook", where the fact that they have already branched out enough is part of their true self.
If you're worried about someone jumping playbooks and just amassing moves, think about the rule about dropping what belongs to your old life. Out of those eight playbook moves you've collected over your different careers, none of them belong to those parts of your past? None of them represents stuff you left behind when you switched playbooks? If no, why did you change playbooks? In practice I don't think this is going to be a problem.