Re: We're a Big MyISAM Shop, But...
This horrible bug (still open!) is one reason to start with:
http://bugs.mysql.com/bug.php?id=57118
After that, you have to realize that to be able to gain anything from InnoDB if you have big databases/tables, you'll also need to reserve some serious amounts of RAM for the InnoDB specific buffers as well.
Joining between these 2 table types does not prevent you from locking problems as InnoDB tables will have row-level locking while MyISAM tables will still have table locks. The locking of your queries will be essentially be a table-lock as you're as slow as the more restrictive of your 2 table types.
MySQL 8.0 will barely support MyISAM and it was announced it will be deprecated. Already with version 8.0, some features will not be available.
https://www.percona.com/blog/2016/10/11/mysql-8-0-end-myisam/
Rollbacking when mixing InnoDB & MyISAM is not possible.
Referential integrity not being supported by MyISAM, what happens if you mix both ? Well, could end up having partial records in your database (present in one table but not in the other).