So basically the latest WordPress is not compatible with MySQL < 5.5.3 at all? That seems like a pretty major change.
No, it's perfectly compatible. It just updates the database to use utf8mb4 mechanisms on 5.5.3 and later installations.
If you're actually migrating databases around (which is a bad practice to begin with), then they need to be between similar versions. MySQL has a version breakage at that point, and 4 byte character encodings don't work on the lower versions. So, WordPress only uses those on versions that actually support it.
WordPress does not have any built-in ways to move database data around between different sites. So, don't do that, or do it in a way that isn't just using mysql dump files. Do an export/import instead, or use a migration plugin that handles this sort of thing for you.