Following up after a second interview, which may well be the final stage, should allow the interviewer to be reminded why you want the job. After the first stage, a short note thanking them for their time is enough. At second stage you should list 1 or 2 reasons (only, not a massive list) about what you like. For example, I really liked what you had to say about this new project and think my experience in this, will enable me to make it a success. It was also great to hear about the longevity of the team, as I will be committing to my next role for at least five years. You could also share an article or piece of news, relevant to something you discussed.
If you have a pressing question about terms or pay, leave this until the offer stage. You don’t want to look like you only care about how many holiday days you get.