The main problem in migrating old content to new URLs is to maintain ‘age equity’ – an important trust factor that affects rankings – and to prove to Google that we didn’t simply steal the content, which would incur penalties.
Solution: redirect old URLs to new URLs. Don’t just scrap the old domain and waste all that hard-earned trust!
Updating in-content links is also important. Some kind of site-wide search & replace tool can be helpful here. Without this, you risk the common occurrence where sites are linking to pages that were on the old version of their site, or the URL shows it is a test/staging version of their site that’s still being linked to. If you’re not careful in how you update in-content links, and especially if you mis-configure the redirects from the old site, not only can you end up creating lots of error 404s, but even worse, you could end up with infinite redirect loops which are the worst kind of error because they freeze up the browser and take a lot of bandwidth & resources from the browser and from crawlers.
Where content migration is part of a rebranding exercise then you also need to be careful to update all references to the old brand, otherwise you risk looking disorganised and out-of-date, which makes your business look less professional.
An important caveat to note when redirecting content to a new site would be if you are migrating because site A was penalised, that would be one of the rare cases where you might want to avoid redirecting to site B. Repurpose the content (with caution) but don’t transfer the equity in its entirety aided by URL redirection, because part of that equity is a penalty!