A plugin to share users and usermeta tables between independent WordPress installations. This plugin requires CUSTOM_USER_TABLE and CUSTOM_USER_META_TABLE defined into wp-config file

kinsta-share-users.php

<?php
/**
 * @package Kinsta_Share_Users
 * @version 1.0
 */
/*
Plugin Name: Kinsta Share Users
Plugin URI: http://wordpress.org/extend/plugins/#
Description: This is an example plugin for Kinsta blog readers
Author: Carlo Daniele
Version: 1.0
Author URI: http://carlodaniele.it/en/
*/

/** 
 * Duplicate {$pref}_capabilities and {$pref}_user_level rows in {$pref}_usermeta table
 *
 * @param int    $user_id   The user ID.
 * @param string $role      The new role.
 * @param array  $old_roles An array of the user's previous roles.
 * 
 * @link https://developer.wordpress.org/reference/hooks/set_user_role/
 * @link https://codex.wordpress.org/Plugin_API/Action_Reference/set_user_role
 * 
 */
function ksu_save_role( $user_id, $role ) {

	// Site 1
	// Change value if needed
	$prefix_1 = 'first_';
	
	// Site 2 prefix
	// Change value if needed
	$prefix_2 = 'second_';
	
	$caps = get_user_meta( $user_id, $prefix_1 . 'capabilities', true );
	$level = get_user_meta( $user_id, $prefix_1 . 'user_level', true );

	if ( $caps ){
		update_user_meta( $user_id, $prefix_2 . 'capabilities', $caps );
	}

	if ( $level ){
		update_user_meta( $user_id, $prefix_2 . 'user_level', $level );
	}
}

add_action( 'set_user_role', 'ksu_save_role', 10, 2 );

When there are more than two sites to connect, an "or" is performed within the "If"
Ejm:
if ( $caps ){ update_user_meta( $user_id, $prefix_2 . 'capabilities', $caps ); "or" update_user_meta( $user_id, $prefix_3 . 'capabilities', $caps ); "or" update_user_meta( $user_id, $prefix_4 . 'capabilities', $caps ); }