You can run Liquid in javascript and stylesheet files so compiled CDN output can include values from the site (for example a logo URL from site settings). This is not the same as running Liquid in a page template. Several objects and methods that work on pages do not work here.
@imported stylesheet partials that contain Liquid compile in their own context. They do not share Liquid variables with the parent unless you use {% include %}.
entity or request.{% set_cookie %}).Use {% include javascript|stylesheet|template "path" %}. If you omit the type, it defaults to the current object type. Paths are relative to the current file unless they start with /.
Objects you load with Liquid from javascript or CSS do not mark the parent file dirty. If you change a logo in site settings, a menu, a gallery, or another object referenced only from Liquid in JS/CSS, republish the javascript or stylesheet so the CDN copy updates.