Force WordPress to load the most recent version of the child theme's style.css file.

add-file-mod-time-as-ver-number-to-css.php

<?php
/**
 * Filter all CSS URLs being loaded on the page.
 * Add the last modified time (in Unix Time, i.e. a large integer)
 * to the end of the URL.
 *
 * This forces the browser to load the most recent version of this file. 
 * e.g. http://example.com/wp-content/my-child-theme/style.css
 * becomes
 * http://example.com/wp-content/my-child-theme/style.css?ver=1533080875
 *
 * Quick Setup:
 * 1. Navigate (or create) the mu-plugins directory
 * 2. From the command line run
 * curl -O https://gist.githubusercontent.com/salcode/b627efd3122155e657688914dfaa298f/raw/add-file-mod-time-as-ver-number-to-css.php
 */
add_filter( 'style_loader_src', function( $src, $handle ) {
	// All of my CSS changes are made in a child-theme.
	if ( 'child-theme' !== $handle ) {
		// This is not the child-theme CSS file, make no changes.
		return $src;
	}
	// Use the default location for the child theme CSS file.
	$file_path = get_stylesheet_directory() . '/style.css';
	if ( ! is_file( $file_path ) ) {
		// No file exists in the default child theme CSS file location.
		// Make no changes.
		return $src;
	}
	$last_modified = filemtime( $file_path );
	$src = add_query_arg( 'ver', intval( $last_modified ), $src );
	return $src;
}, 9999, 2 );