updating associated records' tsvectors when the primary record is updated
Make an after-update trigger on "customer" which does a dummy update on "project", which will then fire the trigger on project to do its thing:

CREATE OR REPLACE FUNCTION customer_depend_trigger() RETURNS trigger AS $$
    BEGIN
        update project set customer_id=NEW.id where customer_id=NEW.id;
        RETURN NEW;
    END;
$$ LANGUAGE plpgsql;

create trigger foobar after update ON customer 
    for each row execute procedure customer_depend_trigger() ;