UPDATE: As @tank answers below, Chrome version 77 added "Copy Styles" when you right-click on an element in the devtools inspector.

Using Javascript worked best for me. Here's how I did it:

1. Open Chrome DevTools console.

2. Paste this dumpCSSText function from this stack overflow answer into the console, and hit Enter:

   function dumpCSSText(element){
     var s = '';
     var o = getComputedStyle(element);
     for(var i = 0; i < o.length; i++){
       s+=o[i] + ':' + o.getPropertyValue(o[i])+';';
     }
     return s;
   }
   

3. When using Chrome, you can inspect an element and access it in the console with the $0 variable. Chrome also has a copy command, so use this command to copy ALL the css of the inspected element:

   copy(dumpCSSText($0));
   

4. Paste your CSS wherever you like!

For chrome you can use Quick Source Viewer to see all CSS, whether in the body or the head, whether in the served HTML or injected at runtime.

It is better than the dev tools as you don't have to go looking for the style tags, and saving them is just a right click on the link.

I just found the answer to my question on this post Inspect webkit-input-placeholder with developer tools

What I need to do is to enable 'Show user agent shadow DOM' in the Settings panel of Chrome Developer Tools. Now the properties are visible.