Likely the best thing to do is to migrate your content to your new.domains and do a 301 redirect (permanent) from old-domains.TLD. It’s safe and maintains all of your Google love. Google is completely geared up for this scenario and it is documented on their site.
See this checklist for a complete guide to migrating your domain.
Search engines don’t count framed content as duplicate content because the URL of the frame tells the search engine the source, so an iFrame won’t hurt their SEO. There may be security issues if the customer transitions to a secure page from an insecure one. A search engine will not attribute content inside the frame to the framing domain (yournew.domains). If it was that easy to hurt someone’s SEO, a competitor could set up a site framing your site. If you are paranoid, see my blurb below on the canonical URL tag.
If you don’t want to switch to yournew.domains, you could just do a 302 redirect (temporary redirect) or create a CNAME in DNS that points yournew.domains at your main domain.
Google is good at recognizing this use case but to the best way to avoid affecting your Google ranking in this scenario, you need to add a canonical URL tag to any pages that are linked to by yournew.domains (e.g. the home page) in the header like this:…
This tells Google where you want all of your ranking to be attributed.