Changing nameservers without breaking email

Moving the website is one thing; keeping email working during the move is another.

Before changing nameservers, which records do you check? MX records are the obvious ones, but SPF, DKIM, DMARC and verification records for other services can be easy to miss.

Do you copy the whole DNS zone first, or rebuild it record by record? Any advice for checking the new setup before making the switch?