Migration is exactly where pretty promises do the most damage. Below is the honest version, built from sites we've actually migrated.
What moves
Copy and images are the content, and content survives any platform change. The domain moves: it's redirected with a DNS record, and a certificate is issued on the new side. The structure — sections, categories, the shape of the site — can be reproduced as it was: the engine is flexible about hierarchy.
What doesn't move by itself
Content gets migrated by hand — yours or a studio's. There's no universal importer that swallows an arbitrary site whole, and any platform that promises otherwise is describing a narrow special case. For a site with a few dozen pages, that's an evening's work; for a library with hundreds, it's a project whose cost is better estimated in advance.
Custom code from the old site — someone's plugin, a nonstandard calculator — doesn't move. You'll have to rebuild it or drop it; and honestly, most of that kind of code can usually be dropped.
The main thing that determines whether the move is painful: old addresses
If page addresses change and the old ones simply stop working, you throw away every bit of accumulated search visibility. The unglamorous, mandatory fix: a permanent redirect from every old address to its new one. This has to be planned before the switch, page by page — not after traffic has already dropped.
Where there's no matching page — content you decided not to bring over — send the visitor to the closest section by meaning, not to the homepage. A redirect to the homepage is a polite way of losing someone.
What to expect afterward
A short dip while search engines re-crawl the site and link up the addresses is normal and passes on its own. What doesn't pass on its own is a move done without redirects. That's a permanent loss, and no amount of later optimization buys it back.
If you'd rather not do it yourself
Migration is exactly the kind of work partner studios take on: the technical part, the redirect map, and content migration. Their pricing is on the partners page. If you're migrating a small site yourself, the order is: reproduce the structure, migrate the content, build the redirect map, and only then switch the domain. In exactly that order.