I asked a question and got this reply which helped.
UPDATE TABLE_A a JOIN TABLE_B b
ON a.join_col = b.join_col AND a.column_a = b.column_b
SET a.c
An alternative General Plan, which I'm only adding as an independent Answer because the blasted "comment on an answer" won't take newlines without posting the entire edit, even though it isn't finished yet.
UPDATE table A
JOIN table B ON {join fields}
JOIN table C ON {join fields}
JOIN {as many tables as you need}
SET A.column = {expression}
Example:
UPDATE person P
JOIN address A ON P.home_address_id = A.id
JOIN city C ON A.city_id = C.id
SET P.home_zip = C.zipcode;