Okay, now I realize that I’ve misunderstood both ramf and davidhc, so I’ll attempt clarification of my comments.
Davidhc, you are correct that changing from one theme to another does indeed present different home page layouts (i.e., “generic homepage”), now labelled “Blog Home” within the Full Site Editor (i.e., the “Editor”). This is by design and is normal, and [presumably] done for marketing purposes to promote customer usage of new themes. (Note: website builders are WordPress’ customers; customers being the users of the product.)
And as noted by Jessica Lyschik, “migrat[ing]” (i.e., exporting then importing) templates is not directly possible from the Editor at this time, and she provides a work-around using a custom page (in lieu of using the Blog Home template).
Although patterns may be exported as JSON files, and then imported as a new pattern, preliminary testing indicates that any “template parts” within the pattern are not imported with the pattern; that is, [apparently] template parts are stripped off upon import. Hence, exporting the Blog Home template as a custom pattern is achievable, but upon importing into the target theme, the template parts (i.e., “Header,” “Footer” template parts) are stripped off.
As such, remedial re-building of “the look and feel” of template parts is necessary. (However, building “look and feel” and “boilerplate content” into export/import patterns supports transferring customization across block themes, and may mitigate this effort somewhat.)
The upshot is that Jessica Lyschik’s suggestion may be your best alternative at this time.