That's by design as the system is designed to send emails to the parties that are connected to the case (i.e. the generic customer) not the lead.
Now you could solve this with custom code (when an email is created if the regarding record is a case and the customer on the case is generic customer set the email address to the one on the lead) but that's a mess and really doesn't do what you want to do here.
I would however, recommend looking at the process and see if the following is acceptable:-
1) when a lead is created create a contact record for the customer but add and set a "is lead" flag to the contact record.
1b) update the lead record so that the newly created contact record is recorded within the lead
2) use that contact record as the basis of the case record.
This has the following advantages
1) you can do the above via a workflow (so no code required)
2) it means that were this lead to become a customer you will have the complete history
3) things will work as expected so when a reply email arrived dynamics will attach it to the appropriate case.
And I do know that it will result in more contact records being created by the "is a lead" field will allow you to filter them out of any contact based reported.
