I have faced the very same problem many, many times. In the meantime I have learned that one can force browser to clear it'sits cache only by versioning assets loading. So you need to find where your template loads it'sits CSS files, and to add at the very end of loading command some type of versioning string, something like "?v=xxx". Once the browser caches previous version string, you can easily force it to reload that asset by simply changing version number by increasing it.
That is only "best practice" I know of. There is no system plugin so far which can automate this procedure, so you will have to do it manually by editing corresponding template's .php file which loads it's assets (usually some kind of index.php).
The very same trick can be used for reloading .js files too, but they are not part of your question and concern.