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Is it easy to update a listing in DMOZ if you already have one, or should I convince them to continue to use their old domain that already has that beloved link in DMOZ?
There is very, very, very rarely a good reason to change domain names; there is ALWAYS some downside, and ALWAYS a risk of serious effects on business - short term if well managed, often longer.
You are GUARANTEED to lose many incoming links, many directory entries; however thorough your seo recovery, there will be damage in the serps, therefore in referrals, therefore in business.
Read up on 301 redirects and domain name change - but be sure to warn them of the risks, or you may be blamed for mismanagement when it was simply a bad decision.
1) Make sure that it it obvious that the data is not at the old place anymore, but at the new one. The ideal solution (not only regarding ODP) would be to make the old domain redirect to the new one (Having the same content in two places is not a good idea anyway).
2) If that is ready (and not earlier) file an Update URL request for every listing.
From an SEO point of view (nothing to do with DMOZ) also consider the downside of changing from an established URL to a new URL if, as some people believe, age of site factors into SERPS. You may also be sandboxed on a brand new domain (not appear in search for x months).