In more detail
Google's guide to site moves with URL changes sets out the essentials:
- Map every old URL to its new equivalent before launch, and implement permanent server-side 301 redirects
- Update internal links, canonical tags and hreflang to point to the new URLs
- Submit new sitemaps and use Search Console's Change of Address tool when moving domains
- Keep redirects in place for at least a year, and keep control of the old domain
- Monitor crawling, indexing and traffic closely after launch
Google notes that a medium-sized site can take a few weeks for most pages to move in its index, and larger sites take longer. Some fluctuation is normal while signals transfer.
Common causes of failed migrations include missing or wrong redirects, redirecting everything to the home page, removing content that ranked, leaving staging rules such as noindex or Disallow: / in place, changing many things at once (domain, URLs and content together) and launching without benchmarks to compare against.
Redesigns count too. Even on the same domain, new templates can change titles, headings, internal links and content in ways that affect rankings.
Why it matters when hiring an agency
Involve SEO specialists before a migration, not after. Google's own hiring advice says a great time to hire is when you're considering a site redesign ("the earlier, the better"), so that the new site is search-friendly from the start (source).
Ask agencies or freelancers for examples of migrations they've handled, how they build and test redirect maps, what they check on launch day and how long they'll monitor afterwards. Migration support is usually a fixed-fee project.