A couple of things to be aware of if using the WordPress general Export/Import tool at Tools > Export
If you already have a saved template or template part with the same name (slug) on the new site, it will not be updated.
The general WordPress importer was never designed to take into account the template/part structure of full site editing, so if it encounters e.g. an already existing user saved template part with the ‘header’ slug it will simply import your new (exported from old site) template part and save it with the slug ‘header-2’ or ‘header-3’ and so on.
This different slug will then not match with the template part that is being called from within your templates, and you would then need to manually change all instances of <!-- wp:template-part {"slug":"header"} /-->
in your templates to <!-- wp:template-part {"slug":"header-2"} /-->
for example.
It also does not take into account the theme of the saved templates/parts. This may not be an issue if you only ever had one theme active, but the more themes that have ever been active in your site, the more complicated it gets.
Things can get real complicated pretty quickly if you have a lot of edited templates and/or parts that you want to import to another site.