Tentar
UPDATE `table_a` AS a
INNER JOIN `table_b` AS b ON ( a.`b_id` = b.`id` )
SET a.`field1` = b.`field1`, a.`field2` = b.`field2`
UPDATE `table_a` AS a
INNER JOIN `table_b` AS b ON ( a.`b_id` = b.`id` )
SET a.`field1` = b.`field1`, a.`field2` = b.`field2`
