If a colon is in a user search query in the WP Users dashboard or the PMPro Members List, try to speed up the query.

wordpress_colon_searches.php

<?php
/**
 * Functions affecting admins and the admin dashboard.
 */
/**
 * Return array of fields found in the wp_users table.
 * This function used in the others below to determine if a key entered in a search like "email:address@domain.com"
 * is referencing the user_email column of the wp_users table or a unique meta key.
 */
function my_pmpro_get_user_table_columns() {
	return array( 'login', 'nicename', 'email', 'url', 'display_name' );
}
/**
 * Speed up searching by email in the PMPro Members List.
 * Use a colon in your search to denote the key.
 * email:search will search the user_email column of the wp_users table.
 * meta_key:search will search the wp_usermeta table.
 */
function pmpro_members_list_sql_colon_search( $sqlQuery ) {
	global $wpdb;
	
	if( !empty( $_REQUEST['s'] ) ) {
		$s = sanitize_text_field( $_REQUEST['s'] );
	} else {
		$s = null;
	}
	
	if( !empty( $s ) && strpos( $s, ':' ) !== false ) {
		//user specified key
		$parts = explode( ':', $s );
		$key = $parts[0];
		$s = $parts[1];
		//some vars for the search
		$l = intval( $_REQUEST['l'] );
		
		if(isset($_REQUEST['pn']))
			$pn = intval( $_REQUEST['pn'] );
		else
			$pn = 1;
			
		if(isset($_REQUEST['limit']))
			$limit = intval( $_REQUEST['limit'] );
		else
			$limit = 15;
		
		$end = $pn * $limit;
		$start = $end - $limit;	
		
		//search only on this one meta key
		$sqlQuery = "SELECT SQL_CALC_FOUND_ROWS u.ID, u.user_login, u.user_email, UNIX_TIMESTAMP(u.user_registered) as joindate, mu.membership_id, mu.initial_payment, mu.billing_amount, mu.cycle_period, mu.cycle_number, mu.billing_limit, mu.trial_amount, mu.trial_limit, UNIX_TIMESTAMP(mu.startdate) as startdate, UNIX_TIMESTAMP(mu.enddate) as enddate, m.name as membership FROM $wpdb->users u LEFT JOIN $wpdb->usermeta um ON u.ID = um.user_id LEFT JOIN $wpdb->pmpro_memberships_users mu ON u.ID = mu.user_id LEFT JOIN $wpdb->pmpro_membership_levels m ON mu.membership_id = m.id ";
		
		if($l == "oldmembers")
			$sqlQuery .= " LEFT JOIN $wpdb->pmpro_memberships_users mu2 ON u.ID = mu2.user_id AND mu2.status = 'active' ";
		
		//this is the part of the query that is changed
		//figure out if it's a user column or user meta search
		if( in_array( $key, my_pmpro_get_user_table_columns() ) ) {
			$key_column = 'u.user_' . esc_sql($key);
			$sqlQuery .= " WHERE $key_column LIKE '%" . esc_sql($s) . "%' ";
		} elseif ( $key == 'discount' || $key == 'discount_code' || $key == 'dc' ) {
			$user_ids = $wpdb->get_col( "SELECT dcu.user_id FROM $wpdb->pmpro_discount_codes_uses dcu LEFT JOIN $wpdb->pmpro_discount_codes dc ON dcu.code_id = dc.id WHERE dc.code = '" . esc_sql($s) . "'" );
			$sqlQuery .= " WHERE u.ID IN(" . implode(",", $user_ids) . ") ";
		} else {
			$user_ids = $wpdb->get_col( "SELECT user_id FROM $wpdb->usermeta WHERE meta_key = '" . esc_sql($key) . "' AND meta_value lIKE '%" . esc_sql($s) . "%'" );
			$sqlQuery .= " WHERE u.ID IN(" . implode(",", $user_ids) . ") ";
		}
		//---
	
		if($l == "oldmembers")
			$sqlQuery .= " AND mu.status = 'inactive' AND mu2.status IS NULL ";
		elseif($l)
			$sqlQuery .= " AND mu.status = 'active' AND mu.membership_id = '" . esc_sql( $l ) . "' ";					
		else
			$sqlQuery .= " AND mu.status = 'active' ";			
		
		$sqlQuery .= "GROUP BY u.ID ";
		
		if($l == "oldmembers")
			$sqlQuery .= "ORDER BY enddate DESC ";
		else
			$sqlQuery .= "ORDER BY u.user_registered DESC ";
		
		$sqlQuery .= "LIMIT $start, $limit";
	}
	
	return $sqlQuery;
}
add_filter( 'pmpro_members_list_sql', 'pmpro_members_list_sql_colon_search' );
/**
 * Speed up search by email or user meta in the WP Users search in the dashboard.
 * Use a colon in your search to denote the key.
 * email:search will search the user_email column of the wp_users table.
 * meta_key:search will search the wp_usermeta table.
 */
function pre_user_query_colon_search( $user_query )
{
	// Make sure this is only applied to user search
	if ( $user_query->query_vars['search'] ){
		$search = trim( $user_query->query_vars['search'], '*' );
		if ( $_REQUEST['s'] == $search ){
			global $wpdb;
 
			//check for colons
			if( !empty( $search ) && strpos( $search, ':' ) !== false ) {
				//user specified key
				$parts = explode( ':', $search );
				$key = $parts[0];
				$search = $parts[1];
				if( in_array( $key, my_pmpro_get_user_table_columns() ) ) {
					$key = 'user_' . $key;
					$user_ids = $wpdb->get_col( "SELECT ID FROM $wpdb->users WHERE " . esc_sql($key) . " LIKE '%" . esc_sql($search) . "%'");
				} else {
					$user_ids = $wpdb->get_col( "SELECT user_id FROM $wpdb->usermeta WHERE meta_key = '" . esc_sql($key) . "' AND meta_value lIKE '%" . esc_sql($search) . "%'" );
				}
			} elseif ( function_exists( 'wp_is_large_user_count' ) && wp_is_large_user_count() ) {
				// Just search the users table fields.
				$user_ids = $wpdb->get_col( "SELECT ID FROM $wpdb->users WHERE user_login LIKE '%" . esc_sql( $search ) . "%' OR user_nicename LIKE '%" . esc_sql( $search ) . "%' OR user_email LIKE '%" . esc_sql( $search ) . "%'" );
			}
			
			if(!empty($user_ids))
			{
				$user_query->query_where = "WHERE 1=1 AND ID IN(" . implode(",", $user_ids) . ") ";
			}	
		}
	}
}
add_action( 'pre_user_query', 'pre_user_query_colon_search', 20 );

This recipe is included in the blog post on "Speed Up Member and User Search in Your WordPress Site" at Paid Memberships Pro here: https://www.paidmembershipspro.com/speed-up-member-and-user-search-in-your-wordpress-site/