Add body classes for current page and all ancestors

functions.php

/**
 * Add classes for pages and page parents
 */
function prefix_add_specific_body_classes($classes) {
	
	// Bail if we're not on a page
    if (is_page()) {
		
        global $post;
        
        // If we *do* have an ancestors list, process it
        // http://codex.wordpress.org/Function_Reference/get_post_ancestors
        if ( $parents = get_post_ancestors( $post->ID ) ) {
			foreach ( (array) $parents as $parent ) {

                // As the array contains IDs only, we need to get each page
                if ( $page = get_page( $parent ) ) {
					
					// Add the current ancestor to the body class array
					$classes[] = "{$page->post_type}-{$page->post_name}";
					
                }
            }
        }
 
        // Add the current page to our body class array
        $classes[] = "{$post->post_type}-{$post->post_name}";
    }
    
    return $classes;
}
// Add a filter for WP
add_filter( 'body_class', 'prefix_add_specific_body_classes' );