WP Customizer

Did you read the core code for the menus?
Like any control, the preview is refreshed on value change. The default is to refresh the entire page, but you can change it to do partial refresh.
The function is called bind and it means that there is data that when it changes, the supplied function is called.

Yeah, I tried to find but didn’t. So I desided to text you to ask about howto.
For now the menus are not refreshing the entire page but only the location I am changing in. Can you advise how to change it with not breaking the work of menus to make it refresh the page on any(sorting, location checkboxes, renaming) change ?

You can look at
https://core.trac.www.ads-software.com/browser/tags/5.2.3/src/wp-includes/customize/class-wp-customize-nav-menu-setting.php
and
https://core.trac.www.ads-software.com/browser/tags/5.2.3/src/js/_enqueues/wp/customize/preview-nav-menus.js

You also can dynamically change the transport method from postMessage to refresh.