Is it possible to update a field value in one table based on data inserted into various other tables?
Consider switching the problem around...

Have a TIMESTAMP with suitable DEFAULT in each table.  Have it automatically set on UPDATE or INSERT.

Then, during the SELECT, JOIN the 4 tables and get GREATEST(t1.ts, t2.ts, ...) to find out the latest TIMESTAMP of the 4.

(I have been bitten by TIMESTAMP DEFAULTs -- they get updated when I am not expecting it.  I would rather take control and change/set the ts in the code.)