In my case, the accepted solution was just too slow. For a table with 180K rows the rate of updates was about 10 rows per second. This is with the indexes on the join elements.

I finally resolved my issue using a procedure:

CREATE DEFINER=`my_procedure`@`%` PROCEDURE `rescue`()
BEGIN
    declare str VARCHAR(255) default '';
    DECLARE n INT DEFAULT 0;
    DECLARE i INT DEFAULT 0;
    DECLARE cur_name VARCHAR(45) DEFAULT '';
    DECLARE cur_value VARCHAR(10000) DEFAULT '';
    SELECT COUNT(*) FROM tableA INTO n;
    SET i=0;
    WHILE i<n DO 
      SELECT namea,valuea FROM tableA limit i,1 INTO cur_name,cur_value;
      UPDATE tableB SET nameb=cur_name where valueb=cur_value;
      SET i = i + 1;
    END WHILE;

END

I hope it will help someone in the future like it helped me