Changing to InnoDB increases iostat
Disclaimer: This is still very broad and could be anything, so this answer is only some potential reasons for increased IO activity

My first inclination to your question But how can it increase continuously? when you mentioned that the load is 'write-heavy' is that you have autocommit enabled. If you are doing a lot of inserts, this might be your issue (emphasis mine):


  When importing data into InnoDB, turn off autocommit mode, because it performs a log flush to disk for every insert. To disable autocommit during your import operation, surround it with SET autocommit and COMMIT statements:

SET autocommit=0;
... SQL import statements ...
COMMIT;

  
  [src]


If that doesn't help, another major source of IO in a write-heavy environment are your indexes. Make sure you are not over-indexing (have unused indexes). 

Also, InnoDB will write the primary key data to the index file for each secondary index as a way to identify them. So if you have a lot of natural or composite primary keys in tables that have secondary indexes, that could be a lot of extra data that is being written.