This might help: https://www.joomla51.com/forum/transition-template-to-another-site#reply-46464
you can see Oliver's notes there. My company hasn't used the Ashley template so I am not sure what you can expect but for sure you should duplicate the site and perform the update on that. We don't usually have problems with the themes, but have had a few issues with the J51 modules.

Our process: duplicate, uninstall all J51 extensions we are not using. Start the J4 update process (fixing all the issues that appear. But Don't uninstall the J51 template or the J51 extensions you use.) . Clear the Joomla cache, click on check-in and click to to the database fix. Then run the update (the site will appear broken) then reinstall the J4 J51 theme and used J51 extensions. Delete out the HTML folder (templates/j51_colette/html) and reinstall the J4 template again. Then open the template on the backend and save and close. Open each of the J51 extensions and save and close. Then test the site.

I hope that helps!
PS not J51 related but if you use Akeeba we also uninstall that before the J4 update and reinstall it after.