Following scripts adds two SQL Procedures to move a WP network from main domain to sub directory ( here "/blogs" ).

wp-network-options-table-update.sql

/*

Following scripts adds two SQL Procedures to move a WP network from main domain to sub directory ( here "/blogs" ).

run_command: to execute a sql query
update_options_tables: loop through all the options table and update the home and siteurl options
script at the end: updates the wp_blogs and wp_site table plus class the update_options_tables procedure

*/

DROP PROCEDURE `run_command`;
CREATE DEFINER=`%`@`%` PROCEDURE `run_command`
(IN `cmd` VARCHAR(255)) 
NOT DETERMINISTIC CONTAINS SQL SQL SECURITY DEFINER BEGIN
  set @command = cmd;
  PREPARE stmt FROM @command;
  EXECUTE stmt;
  DEALLOCATE PREPARE stmt;
END

DROP PROCEDURE `update_options_tables`;
CREATE DEFINER=`%`@`%` PROCEDURE `edit_sub_tables`
(IN `old_url` VARCHAR(255), IN `new_url` VARCHAR(255)) 
NOT DETERMINISTIC CONTAINS SQL SQL SECURITY DEFINER BEGIN

  DECLARE done INT DEFAULT FALSE;
  DECLARE current_table CHAR(32);

  DECLARE sub_site_tables CURSOR FOR SELECT distinct table_name FROM information_schema.tables WHERE table_name like 'wp_%_options';
  
  -- you can use "in ('wp_1_options','wp_2_options','wp_3_options','wp_4_options');" above as well
  
  DECLARE CONTINUE HANDLER FOR NOT FOUND SET done = TRUE;
  OPEN sub_site_tables;

  read_loop: LOOP
    FETCH sub_site_tables INTO current_table;
    IF done THEN
      LEAVE read_loop;
    END IF;

    SET @sql_text = concat('update ', current_table,' set option_value = replace(option_value, "',old_url,'","', new_url,'") where option_name IN( "siteurl","home") ');
    CALL run_command(@sql_text);
  END LOOP;
  CLOSE sub_site_tables;
END



UPDATE `wp_blogs` SET `domain` = 'new_site.com', `path`= concat('/blogs',path);
UPDATE `wp_site` SET `domain` = 'new_site.com', `path`= concat('/blogs',path);

call update_options_tables('old_site.com',  'new_site.com/blogs');