child-theme-switcher.php

<?php

/**
 * @author AppPresser
 * 
 * Lets you use different child themes for multisite.
 * 
 * Needs to added as a mu-plugin. So create a new file and
 * place it in the wp-content/mu-plugins/ folder.
 * 
 * Name your child theme folders using the blog id:
 * 
 * app-theme-14, but remove the ion-ap3-child theme.
 */
function my_app_child_theme( $theme_slug ) {
	
	$new_theme_slug = 'app-theme-' . get_current_blog_id();

	$child_theme = wp_get_theme( $new_theme_slug );

	if ( $child_theme->exists() ) {
		$theme_slug = $new_theme_slug;
	}

	return $theme_slug;

}
add_filter( 'appp_theme', 'my_app_child_theme' );