Hi @juliaset
When pulling a Kinsta site (live or staging), it only pulls the site files and database. Any custom Nginx configurations are not pulled over to DevKinsta.
My guess is the Devkinsta Pull action isn’t able to fully replace all entries of your Kinsta Dev domain with the Devkinsta .local domain. The site uses the Elementor plugin, which seems to use an uninitialized class to manage the domain entries. If there are still Kinsta Dev domain entries left, this will still trigger the password authentication.
I suggest running a separate Search and Replace in your DevKinsta local site within Elemetor using the following:
Search: yourKinstaDomain.com (replace with your actual Kinsta domain)
Replace with: yourDevKinstadomain.local (replace with your actual Devkinsta domain)
Ref: Will Elementor Work If I Change The Domain Of My Website? » Elementor
I hope it helps; let us know how it goes.