Hello

is it possible to transfer a mobile layout from a local installation of a website to the online version - for example via the database entries of the responsivizer Component, or what is necessary to do this without having to make a new layout in the backend.

Hello,

it depends on what you have done on the local version for the mobile layout. If you worked mainly on css, you can copy the local css files of Responsivizer to the remote host. If you have created some modules for mobile the database table of Joomla that holds modules is involved.
About the settings of Responsivizer, they are stored in the #__extensions table of Joomla.
If you need to transfer all setup at once an interesting solution would be using Akeeba backup to clone the site on the remote host.

Best regards

John

so I will test it - copying the css files and import the settings from the database tables -

Thank you